А.С. Мазманишвили. Математическая статистика. Учебное

. . 7.080201 ""#" 7.080202 "$ " % - ' () "*$"" "" 2003 22.171 12 519.2 #$#%&#%'( : .. *+%,-., *{ '#1%. %2, 34., ,-. %$-%5%(. 2%-#,-'#' *-65#'%-7.. 8-., *{ '#1%. %2, 34., ,+ 8,2*,'#%%+ *#9-+ :#5#&%*:%8 '%,3' , @@@@ """"" 12 : . / &9%-<-5- .. | : "", 2003. | 217 ,. | 2,,. +&. ISBN 966{593{270{5 $+ " ". , : , / 0 , / 0. $ , 12/ "$ ". 3 #i{/, i%{/i/ i i/ i ii, % #/i. $i+ " ". i i ii i: i ii, /ii ' , / 'i. $ i, 2 1 ii1 "$ ". 3 #i{/, i%{/i/ i i/ i ii, % #/i. This textbook contains tasks on Mathematical Statistics sorted according to the topics of the course: theoretical information that is necessary for solving the tasks, examples of the solutions. This textbook is created for students on Applied Mathematics. It can also be useful for students on Mathematics, Physics, Engineering and Economy. 5. 55. ?5. 20. -?5-8. 42 %&. 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' 1.1 5+ ,5#*2HO#8 ,3#*#5#%-+ 45 3 92:,. ?2-, 3*%%. 98&-%# & *#%, 3,'-' *-,#'%(. -$-%%(. +*. 40 41 42 40 40 39 40 41 39 41 40 42 41 40 42 41 42 41 40 38 44 39 42 41 42 39 41 37 43 41 44 40 42 41 39 41 42 43 42 41 43 43 38 38 43 20 () 5+ 3,'#%-+ -$-%%8 +* &5-A%(# &%A#%-+ 3-&% ,358#9 3+*# -1 &,'%-+. %A'#5% -$-%%(. +* 3-%-9#' -* : &9# 37 38 39 40 41 42 43 44 R,'' 1 3 5 8 12 9 5 2 ' 1.2 & 3'-- *#'5#. ?(5 '?% 400, ,3#*#5#%-# '(1 3 &9#2 *#',+ ,5#*2HO#. '?5-$#. : &9# *#'5-, 7,95{8,00 8,00{8,05 8,05{8,10 R-,5 *#'5#. 12 28 132 &9# *#'5-, 8,10{8,15 8,15{8,20 8,20{8,25 R-,5 *#'5#. 150 62 16 .'- <-?2 (?- 3- 3#*#5#%-- ,#*%#8. () %A5 %1*-9 (?A%2H *-,3#,-H 2: 2 = 0 00272. D' *#' *5+ <-?- (?- 3- 3#*#5#%-- ,#*%#8 q p = 0= n ; 1 = 0 00272=(400 ; 1) = 0 002611: q 0 00272=400 = 0,002608, ' 3- '-1 ?5<-1 ?T#91 (?- n p 9:% <-?2 (?- %1*-' 3'-A#,- 3 4925# s = 0= n. ' 1.3 #*,'-' (?2 -& n %?5H*#%-. -*# '?5-$( A,'', -,35&2+ 7 -%'#5 8233--. 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A#,'# 3#8 -%'#5 2*?% &+' 10{12 - ' *5## *5+ ,5#*2HO-1 -%'#5. ?&%A-9 : i | %9# -%'#5U î | 8%-$( i-8 -%'#5U xi | ,##*-%( -%'#5U ni | P A,''(U mi = ij=1 nj | %35#%%(# A,''(U pi = nPi=n | '%,-'#5%(# A,''( (A,','-)U Fi = ij=1 pj | %35#%%(# A,','-. 21 #&25''# 352A#9 ,5#*2HO2H '?5-$2 : i î xi ni mi 1 10{12 11 2 2 2 12{14 13 4 6 3 14{16 15 8 14 4 16{18 17 12 26 5 18{20 19 15 41 6 20{22 21 11 52 7 22{24 23 3 55 pi 0,0364 0,0727 0,1455 0,2182 0,2909 0,1818 0,0545 Fi 0,0364 0,1091 0,2546 0,4728 0,7637 0,9455 1,0000 ' 1.4 5+ 3#*#5#%-+ #3,'- %-'- 3#*#%( -,3('%-+ n = 1000 3?, *<-# ,5#*2HO-# #&25''(: #3,' %-'-, ! 180{190 190{200 200{210 210{220 R-,5 3?, ni 50 90 150 280 #3,' %-'-, ! 220{230 230{240 240{250 R-,5 3?, ni 220 120 90 ,'-' 2925+%'%(. +*. () %A5 %1*-9 %35#%%(# A,''( *5+ :*8 -& -%'#5 *%%8 -%'#5%8 -$-%%8 +* : m1 = n1 = 50U m2 = n1 + n2 = 50 + 90 = 140U m3 = 290U m4 = 570U m5 = 790U m6 = 910U m7 = 1000. -9 ?&9, 3,5# %9-- % n = 1000 2925+%'%(. +* *5+ *%%8 ,3#*#5#%-+ -9##' -* : #3,' %-'-, ! 180{190 190{200 200{210 210{220 R-,5 3?, ni 50 90 150 280 Fi = mi=n 0,050 0,140 0,290 0,570 #3,' %-'-, ! 220{230 230{240 240{250 R-,5 3?, ni 220 120 90 Fi = mi=n 0,790 0,910 1,000 ' 1.5 3-,' -*# -$-%%8 - ,''-,'-A#,8 +* (?2 f5 3 7 10 5 5 2 10 7 2 7 7 4 2 4g ,'-' ,''-,'-A#,-. +* - 3#*#5-' &91 (?-. () ?T#9 (?- n = 15. 3+*A-+ 65#9#%'( (?- 3 #5-A-%#, 352A#9 -$-%%(. (%:-%%(.) +* f2 2 2 3 4 4 5 5 5 7 7 7 7 10 10g. 22 &91 (?- ,,'5+#' W = 10 ; 2 = 8. &5-A%(9- &*%%. (?# +5+H',+ 65#9#%'( x1 = 2, x2 = 3, x3 = 4, x4 = 5, x5 = 7, x6 = 10. 1 A,''( ,'#','#%% %( n1 = 3, n2 = 1, n3 = 2, n4 = 3, n5 = 4, n6 = 2. 5#*'#5%, ,''-,'-A#,-. +* -,1*%. (?- 9:% &3-,' -*# ,5#*2HO#. '?5-$( : xi 2 3 4 5 7 10 ni 3 1 2 3 4 2 5+ %'5+ 3-5%,'- %1*-9 Pi ni = 15. ' 1.6 $#5H 3#*#5#%-+ ,#*%#. ,299( 5* ,?##8'#5%. ,,#, -9#HO#. N = 2200 5*A-, 3-&#*#% (?A%# ?,5#*%-# (?#,3'%(. '?) n = 111 5*, '# *5 ,5#*2HO-# #&25''( : 299 5* 10{30 30{50 50{70 70{90 90{110 110{130 R-,5 5* 1 3 10 30 60 7 5&2+, 6'-9- *%%(9-, %.'- *#-'#5%(# 8%-$( *5+ 8#%#5%8 ,#*%#8, '# 9:% ?(5 ?( 8%'-' , #+'%,'H p = 0 96. () _8 -%'#5 ,,'5+#' h = 20. ?&%A-9 ,##*-%2 i-8 -%'#5 A##& xi, i = 1 : : : 6. 5+ 23O#%-+ ,A#' #*#9 ,398'#5%2H ,52A.%2H #5-A-%2 U = (X ; x4)=h = (X ; 80)=20: %A5 %.*#9 (?A%# ,#*%## - (?A%2H *-,3#,-H, *5+ A#8 ,#*#9 (A-,5#%-+ '?5-$2 : i 7%-$( -%'#5 xi mi ui = (xi ; 80)=20 miui miu2i 1 10{30 20 1 {3 {3 9 2 30{50 40 3 {2 {6 12 3 50{70 60 10 {1 {10 10 4 70{90 80 30 0 0 0 5 90{110 100 60 1 60 60 6 110{130 120 7 2 14 28 299 111 55 119 & '?5-$( -9##9 P mu 55 = 0 4955U U = Pi mi i = 111 i i P m u2 U2 = Pi mi i = 119 111 = 1 0721: i i ',H* 2 U2 = U2 ; U = 1 0721 ; 0 2455 = 0 8266 23 A' *#' X = 80 + 20 0 4955 = 89 91U X2 = 202 0 8266 = 330 64: 6'92 <-? 3#*#5#%-+ ,#*%#8 ,,'5+#' s 2 s 330 64 111 X n n ; 1 1 ; N = 111 ; 1 1 ; 2200 = 1 69: a(t) = 0,96 3- t = 2,05 (,9. '?5-$( 42%$-- K35,), ' 3#*#5%+ 38#<%,' " = t = 2,051,69 = 3,46. ', , #+'%,'H p = 0 96 *#-'#5%(9- 8%-$9- *5+ 8#%#5%8 ,#*%#8 ?2*2' X ; " = 89 91 ; 3 46 = 86 45 X + " = 89 91 + 3 46 = 93 37: ' 1.7 :*2H ,'%H -&8'5#%%(1 *#'5#. ,#*%#9 ?(H' *#, %# 2*5#'+HO-1 ,'%*'2 (?). (5 3##% 10 3'-. 3 100 -&*#5-. :*.. '5%#%-+ A-,5 ?%2:#%%(1 ?%%(1 -&*#5-. ' ,#*%#8 3-#*#%( '?5-$# : 9# 3'-1 2 3 4 5 6 7 8 9 10 '5%#%-# ' ,#*%#8 {1 0 1 1 {1 1 0 {2 2 1 ,'-' -$-%%(. +*. .'- (?A%# ,#*%## 25%#%-+ A-,5 ?%%(1 -&*#5-. ' 2,'%5#%%8 - #8 (?A%2H *-,3#,-H. () (?A%(. +* f{2, {1, {1, 0, 0, 1, 1, 1, 1, 2g ,*#:-' 3+' &5-A%(1 &%A#%-.: f{2, {1, 0, 1, 2g. 1 A,','- ,'#','#%% %( f0,1U 0,2U 0,2U 0,4U 0,1g. ?5-$, 3#*,'5+HO+ ,.,' (?A%. ,52A.%. #5-A-%(, ,5#*2HO+ : 9#, i 1 2 3 4 5 %A#%-#, xi {2 {1 0 1 2 R,',', fi 0,1 0,2 0,2 0,4 0,1 (?A%# ,#*%## ,,'5+#' X = 0 2. (?A%+ *-,3#,-+ % 2 = 1 36. ' 1.8 (A-,5-' ,#*%## X - *-,3#,-H X 2 8233-%%. (?( i 1 2 3 8 Bi 134{138 138{142 142{146 C ni 1 3 15 ( i 4 5 6 8 Bi 146{150 150{154 154{158 C ni 18 14 2 24 () ,52A.%. #5-A-%( X, &*%%. (?., *5-% :*8 -& -%'#5 8233-- ,,'5+#' h = 4, &%A#%-# ,##*-%( -%'#5, ,'#AHO#8,+ , %-?5<#. A,''., % d = 148. ?&%A-9 %? fzig | &%A#%-+ ,##*-% i{1 -%'#5 - 3#?&2#9 8233-%%2H (?2 ,5#*2HO-9 ?&9 : ui = xi ;h d = xi ;4 148 i = 1 2 : : : 6 '.#. -,35&2#9 3-#*#%%2H ,52A.%2H #5-A-%2 U. (A-,5#%-+ ,#*#9 '?5-$2 : i zi ui ni niui niu2i 1 136 {3 1 {3 9 2 140 {2 3 {6 12 3 144 {1 15 {15 15 4 148 0 18 0 0 5 152 1 14 14 14 6 156 2 2 4 8 299 { { 53 {6 58 #3# %1*-9 X = hU + d = 4 ;536 + 148 147 55 ;6)2=53 17 30: X 2 = h2 U 2 = 42 58 ; (53 ' 1.9 5+ 3#- ?'( 3899( 3,'#%-+ 8-,'899( ?(5 ??'% 9,,- fxn g, n = 1 2 : : : N , 8*# ?T#9 (?- N = 10000. %A#%-+ 829#%' x ?(5- 352A#%( 3 3-52 xn = n=N , &%A#%-+ 42%$-- y(x) | 3 3-52 yn = sin(xn). -9 ?&9, -,1*%(. 9,,- 3#*,'5+#' ,?. (?2 &%A#%-. #. ,'-' 8-,'8992 9,,- fyng, (? A-,5 %5 %5-& M = 24. () %A#%-+ 42%$-- 8%-A#%(, A#-*%, -%'#59 0 y 1. 6'92 6'9 -%'#5# - ?2*#9 &*' %5( %5-& 8-,'899( <--%. h = 1=M :*(.. 7-,'899 3-#*#% % -, 1.9. . #E -* ,'#','2#' 84-2 35'%,'- %#'. ,52A.%. #5-A-%(, 3*A-%+HO#.,+ &%2 ,-%2,. 2 . ,5- ?( 3##9#%%+ x ?(5 , %9#% ,3#*#5#%%. % -%'#5# (0 x ), ' '-3 8-,'899( ?(5 ?( '-9 :#. ' 1.10 5+ 3#*#5#%-+ ,#*%#8 3$#%' ,(8 ?#5 &#%1 3<#%-$( ?(5 '?% 626 &##%, ?,5#*%-# '(1 3&5, A' (?A%# ,#*%## % 16,8, (?A%+ *-,3#,-+ ,,'-5 4. 25 2000 G(m) 1000 0 0 5 10 15 20 m 25 1.9 | 8+ G(m) # y = sin(x) ( x (0G =2)G H N = 10000G ++ M = 24G + max y = 1) R#92 % #+'%,' p '8, A' ,#*%-. 3$#%' ,(8 ?#5 '5-A#',+ ' 16,8 3 ?,5H'%. #5-A-%# 9#%<# A#9 % 0,2%? () ?T#9 8#%#5%.p,23%,'- %#-&#,'#%. 6'92 <-?2 (?- %1*-9 3 4925# = n ; 1. %<#9 ,52A# n = 626, 2 = 4 -, ,5#*'#5%, p 2= 625 = 2=25 = 0 08: Pr(jX ; Xj < ") a ("=), ' -,9+ #+'%,' p = Pr(jX ; 16 8j < 0 2) a (0 2=0 08) = a(2 5) = 0 99758: ' 1.11 %A#%-+ n %#&-,-9(1 ,52A.%(1 #5-A-% fX1, X2, : : :, Xng, -9#HO-1 *%2 - '2 :# 42%$-H ,3#*#5#%-+ G(x), ,35:#%( 3+*# &,'%-+. 2,' - | &*%%(# $#5(# A-,5 (1 n). .'- ,3#*#5#%-# -8 &%A#%-+ ,%-&2 U - -8 &%A#%-+ ,#12 V 6'9 %:-%%9 +*#. () 5+ '8, A'?( #5-A-% U 335 -%'#5 Zu u + du), %#?1*-9 - *,''A%, A'?( -#-%-?2* ; 1 -& n #5-A-% fX1 X2 : : : Xn g 3-%+5&%A#%-+, 9#%<-# A#9 u, *% &%A#%-# 335 -%'#5 Zu u + du), ,'5%(# n ; 3-%+5- &%A#%-+, %# 9#%<-# A#9 u + du. 26 -9 ?&9, 9( 3-1*-9 ,1#9# 3'#%-+ 3(' , '#9+ %#,9#,'%(9- ,?('-+9-, ?&2HO-9- 35%2H 82332 : A1 = fX < ug A2 = fX u + dug A3 = fu X < u + dug: ,5- ?&%A-' 35'%,' ,3#*#5#%-+ g(u) = dG(u)=du , ' #+'%,'3-#*#%%(1 ,?('-. 3- *%9 3('# %( ,'#','#%% (, 'A%,'H * ?#,%#A% 95(1 (,<-1 3+*) Pr(A1) = G(u) Pr(A2) = 1 ; G(u) Pr(A3) = g(u)du: 8* 352A-9 fU (u) = ( ; 1)n!(!n ; ) G;1 (u) Z1 ; G(u)]n; g(u) | 6' 4925 3#*#5+#' 35'%,' ,3#*#5#%-+ -8 &%A#%-+ +*. %58-A% %1*-9 35'%,' -8 &%A#%-+ V fV (v) = ( ; 1)n!(!n ; ) G;1 (v) Z1 ; G(v)]n; g(v): 5+ %:-%%8 +* ,,9'-9 ,52A.%(# #5-A-%( : %-9#%<## &%A#%-# +* U = 1min fX gU in i %-?5<## &%A#%-# +* V = 1max fX g. in i - = 1 - = 1 352A#9 *5+ 35'%,'- ,3#*#5#%-+ %-9#%<#8 U %-?5<#8 V &%A#%-# ,,9'-#98 +* fU (u) = nZ1 ; G(u)]n;1 g(u) fV (v) = nGn;1 (u) g(v): -#*#9 ':# (:#%-# *5+ 35'%,'- ,3#*#5#%-+ &%,'- 9#:*2 -9 &%A#%-#9 ,#12 - -9 &%A#%-#9 ,%-&2 R = V ; U fR (r) = Z( ; 1)!]n2 !(n ; 2 )! Z1 g(x)g(r + x)G;1(x)ZG(r + x) ; G(x)]n;2 Z1 ; G(r + x)];1 dx: ;1 A,'%9 ,52A# 3- = 1 ',H* 352A#9 35'%,' ,3#*#5#%-+ 3 S = 1max fX g ; 1min fX g in i in i (?# -& n %#&-,-9(1 ,52A.%(1 #5-A-% (% ) Z1 fS (s) = n(n ; 1) ZG(x + s) ; G(x)]g(x)g(s + x) dx: ;1 27 1.8. ) * 1.1 ,3('(5, A2,'-'#5%,' 40 3-#9%-. %%(# 3-#*#%( '?5-$#, '. 3#. ,'# *%( -%'#5( A2,'-'#5%,'- 9-5'1, '. | ,#*%-# 'A- 6'-1 -%'#5 f, '#'#. | A-,5 3-#9%- ni, A2,'-'#5%,' '(1 &5, 6'9 -%'#5#. " 25{75 75{125 125{175 175{225 225{275 50 100 150 200 250 0 0 1 5 8 " 275{325 325{375 375{425 425{475 475{525 f 300 350 400 450 500 ni 6 8 6 2 2 " 525{575 575{625 625{675 675{725 725{775 f 550 600 650 700 750 ni 0 1 1 0 0 f ni ,'-' 693--A#,2H 42%$-H ,3#*#5#%-+ - 8-,'8992 (?-, %.'- ,#*%HH A2,'-'#5%,' 3-#9%- -& 6'. 3'-- - #E ,#*%#(?A%# 25%#%-#. 1.2 '( : ?5H*#%-+ & '5O-%. ( ) ,5H*+%(1 ?&$ *5- ,5#*2HO-# #&25'- 0,030 0,039 0,021 0,030 0,039 0,031 0,042 0,034 0,036 0,030 0,028 0,031 0,042 0,030 0,033 0,024 0,031 0,040 0,031 0,033 0,031 0,027 0,037 0,031 0,031 0,045 0,031 0,034 0,027 0,030 0,048 0,030 0,028 0,036 0,028 0,030 0,033 0,046 0,043 0,030 0,033 0,028 0,031 0,027 0,026 0,031 0,036 0,051 0,034 0,031 0,036 0,034 0,037 0,028 0,030 ,'-' 3 6'-9 *%%(9 -%'#5%(. -$-%%(. +* , %(9-%'#59- (3#(. -%'#5 0,020{0,024, '. 0,024{0,028 - '.*.) - %A#'-' 8-,'8992. 1.3 #*%++ '#93#'2 &*21 ,#%'+?# *21 8*1 (X) - (Y) -&9#+5, '#A#%-# 40 5#'. %%(# 3-#*#%( '?5-$#. X Y X Y X Y X Y X Y 12,0 10,8 13,9 10,1 14,9 13,0 16,0 16,0 18,0 14,0 12,0 11,3 14,2 10,0 14,9 14,2 16,9 12,9 18,0 14,9 12,0 12,0 14,0 10,0 15,1 13,8 17,2 13,9 18,1 16,0 12,0 13,0 14,0 12,0 15,0 16,0 16,9 14,8 18,4 17,8 12,8 10,9 13,9 12,4 15,5 13,9 16,9 15,0 19,2 15,0 13,8 10,0 15,0 11,0 15,9 14,7 17,0 16,0 19,3 16,1 13,1 13,0 14,0 14,8 16,0 13,0 16,8 17,0 20,0 17,0 13,0 13,0 14,0 15,2 15,9 15,0 17,5 16,0 20,1 17,7 28 .'- (?A%(# ,#*%#9#,+A%(# '#93#'2( ?-1 %,#5#%%(1 32%'1 - -1 ,#*%#*'-A%(# 25%#%-+. 1.4 ?,5#*%-# 3&5 ,5#*2HO## ,3#*#5#%-# ,' 8233( H%<#. : ,', R-,5 H%<#. ,', R-,5 H%<#. 143{146 1 167{170 170 146{149 2 170{173 120 149{152 8 173{176 64 152{155 26 176{179 28 155{158 65 179{182 10 158{161 120 182{185 3 161{164 181 185{188 1 164{167 201 188{191 1 ,'-' 2925+%'%(. +* - %A#'-' 2925+'2 - 8-2. 1.5 52A.%+ #5-A-% X %9#% ,3#*#5#% % -%'#5# (0 2), ,52A.%+ #5-A-% Y ,+&% , X ,'%<#%-#9 Y = sin(X). 52A#% (? ?T#99 n &%A#%-. ,52A.%. #5-A-%( Y. -#,'- *-44##%$-5%(. - -%'#85%(. &%( ,3#*#5#%-+ ,52A.%. #5-A-%( Y. ,'-' 8-,'8992 - 2925+'2 (?-. 1.6 ?,5#*%-# 3&5 ,5#*2HO## ,3#*#5#%-# ,' 8233( *#2<# : ,', R-,5 *#2<# ,', R-,5 *#2<# 133{136 1 157{160 150 136{139 2 160{163 109 139{142 18 163{166 54 142{145 36 166{169 18 145{148 85 169{172 8 148{151 140 172{175 1 151{154 211 175{178 0 154{157 181 178{181 0 .'- 9*2, 9#*-%2 - ,#*%## -49#'-A#,# 6'8 ,3#*#5#%-+. 1.7 ?5H*#%-+ & '5O-%. ( ) 9#*%(1 ?&$ *5- ,5#*2HO-# #&25''( : 0,031 0,040 0,049 0,041 0,052 0,044 0,046 0,040 0,038 0,040 0,043 0,034 0,041 0,050 0,041 0,043 0,041 0,037 0,041 0,055 0,041 0,044 0,047 0,040 0,058 0,040 0,038 0,040 0,043 0,056 0,053 0,040 0,043 0,038 0,041 0,037 0,041 0,046 0,061 0,044 0,041 0,046 0,044 0,047 0,038 0,040 0,049 0,041 0,042 0,037 ,'-' 2925+%'%(. +* - %A#'-' 2925+'2 - 8-2 ,''-,'-A#,8 29 ,3#*#5#%-+ 3-#*#%%(1 *%%(1. 1.8 95A%. 4#9# &#8-,'-5- ,#*#%-+ #5-A-%# 2*+ : *., ! 5-A#,' *., ! 5-A#,' 400{600 1 1600{1800 14 600{800 3 1800{2000 12 800{1000 6 2000{2200 10 1000{1200 11 2200{2400 6 1200{1400 15 2400{2600 2 1400{1600 20 2600 - (<# 2 .'- *-,3#,-H, 644-$-#%' -$-- - &91 -$-- ,3#*#5#%-+. 1.9 52A.%+ #5-A-% X %9#% ,3#*#5#% % -%'#5# (0 2), ,52A.%+ #5-A-% Y ,+&% , X ,'%<#%-#9 Y = tg(X). 52A#% (? ?T#99 n &%A#%-. ,52A.%. #5-A-%( Y. -#,'- *-44##%$-5%(. - -%'#85%(. &%( ,3#*#5#%-+ ,52A.%. #5-A-%( Y. ,'-' 8-,'8992 - 2925+'2 (?-. 1.10 R##& :*(. A, -&9#+5, %3+:#%-# ' 65#',#'-. - 6'9 ?(5352A#%( ,5#*2HO-# &%A#%-+ ( V ) : 227 219 215 230 232 223 220 222 218 219 222 221 227 226 226 209 211 215 218 220 216 220 220 221 225 224 212 217 219 220 220 222 218 218 219 222 218 225 224 212 217 219 220 220 ,'-' ,''-,'-A#,# ,3#*#5#%-# - %A#'-' 35-8%. 1.11 7233 ?-'2-#%' 352A-5- ,5#*2HO-# ?55( % 3-E9%(1 6&9#%1: 20 19 22 24 21 18 23 17 20 16 15 23 20 21 24 21 24 21 20 18 19 17 22 20 16 22 18 20 17 21 17 19 22 23 21 25 22 20 19 21 24 22 23 21 29 22 21 19 21 19 20 20 21 18 22 25 21 18 23 20 16 20 23 22 .'- 9*2, 9#*-%2 - ,#*%## -49#'-A#,# 6'8 ,3#*#5#%-+. 1.12 ,'-' *-,#'%(. -$-%%(. +* - %A#'-' 35-8% ,3#*#5#%-+ 8233( ?-'2-#%' 3 A-,52 ?55, 352A#%%(1 -9- % 3-E9%(1 6&9#%1 : 20 19 22 24 21 18 23 17 20 16 15 23 21 24 21 24 21 20 18 17 22 20 16 22 18 20 17 21 17 19 20 19 21 24 22 23 21 23 22 21 19 20 23 22 25 22 23 21 25 22 20 16 20 21 18 18 23 21 19 20 30 1.13 %%(# ? 2:.%,'- :- % &5-A%(1 2A,'1 35+ 3-#*#%( ,5#*2HO#. '?5-$# : :.%,', #/! 9{12 12{15 15{18 18{21 21{24 24{27 5+ 2A,' ' ?O#. 6 12 33 22 19 8 3,#%. 35O*-, % ,'-' 2925+%'%(. +*, %A#'-' 2925+'2 - 8-2. 1.14 ?5H*#%-+ & 3$#%'9 :- 95# 50 *5- #&25''( : 3,86 4,06 3,67 3,97 3,76 3,61 3,96 4,04 3,84 3,94 3,98 3,57 3,87 4,07 3,99 3,69 3,76 3,71 3,94 3,82 4,16 3,76 4,00 3,46 4,08 3,88 4,01 3,93 3,71 3,81 4,02 4,17 3,72 4,09 3,78 4,02 3,73 3,52 3,89 3,92 4,18 4,26 4,03 4,14 3,72 4,33 3,82 4,03 3,62 3,91 ,'-' 3 6'-9 *%%(9 -%'#5%(. -$-%%(. +* , %(9-%'#59- (3#(. -%'#5 3,45{3,55 %, '. -%'#5 3,55{3,65 % - '.*.) -&?&-' #8 84-A#,-. 1.9. ) %(? 1. -# &*A- ,,9'-#' 9'#9'-A#,+ ,''-,'-? 2. R' %&(#',+ 8#%#5%. ,23%,'H, (?A%. ,23%,'H? 3. A#9 &5HA#',+ ,2O%,' (?A%8 9#'*? 4. R' %&(#',+ ,''-,'-A#,-9 &%9 ,3#*#5#%-+ ,52A.%. #5-A-- 5. R' %&(#',+ 693--A#,. 42%$-#. ,3#*#5#%-+ ,52A.%. #5-A-%(? 6. A#9 ,,'-' &5-A-# 9#:*2 693--A#,. 42%$-#. ,3#*#5#%-+ '##'-A#,. (-%'#85%.) 42%$-#. ,3#*#5#%-+ ,52A.%. #5-A-%(? 7. -# ,.,' -9##' 693--A#,+ 42%$-+ ,3#*#5#%-+ ,52A.%. #5-A-%(? 8. ##A-,5-'# ,%%(# -*( 84-, ,52:O-1 *5+ -&?:#%-+ ,''-,'-A#,-1 +*. 31 2. -?5## A,' ,'#AHO-#,+ ,3#*#5#%-+ ,%%(1 ,''-,'-, (A-,5+#9(1 3 (?# -& %95% ,3#*#5#%%. 8#%#5%. ,23%,'-, ,+&%( , %95%(9 &%9 72,,, 899-,3#*#5#%-#9, ,3#*#5#%-#9 1--*' f (2), ,3#*#5#%-#9 'H*#%' - ,3#*#5#%-#9 J-<#-%#*#. - -1 ,,9'#%-- 9( ':# ,'#'-9,+ , 899.-42%$-#. D.5# ;(). &*A1 9'#9'-A#,. ,''-,'-- ':# A,' -,35&2H',+ ,3#*#5#%-+ 598 598-9-% K (), ,3#*#5#%-# '5#'' - *28-#. 2.1. , 52A.%+ #5-A-% X -9##' , 39#'9- m - 2, #,5- #E 35'%,' ,3#*#5#%-+ ,5#*2HO+ : 2! 1 ( x ; m ) fX (x) = p (2:1) exp ; 2 2 2 ?5,'H 3#*#5#%-+ X +5+#',+ ,+ A-,5+ ,, ;1 < x < 1. '2H ,52A.%2H #5-A-%2 #OE 8+', A' . D' ,52A.%+ #5-A-% %-?5## <- 3-9#%+#',+ 3- 3,'#%-,''-,'-A#,-1 9*#5#.. 3'-# *5+ ','- %95%2H ,52A.%2H #5-A-%2 X , 39#'9m - A,' ?&%AH' N (mU ). -+ ,3#*#5#%-+, 3-,(HO+ 35'%,' (2.1), ,-99#'-A% '%,-'#5% 'A- m, '. 35'%,' *,'-8#' 9,-929. & 6'. ,-99#'-- %#3,#*,'#%% ('##', A' 9'#9'-A#,# :-*%-# ,52A.%. #5-A-%( X % MZX] = m: (2:2) -&9#%#%-#9 &%A#%-+ 9'#9'-A#,8 :-*%-+ -+, $#5#, ,9#O#',+ , ,'%2 -&9#%#%-+, p ;16' 3&% % -,. 2.1{2.2. 'A# x = m *,'-8#',+ 9,-929, %(. 2 = 0 3989= . -,3#,-+ %95%. ,52A.%. #5-A-%( X % DZX] = 2: (2:3) -9 ?&9, 39#' 9:% -%'#3#'-' 9#2 ,,#+%-+ ,52A.%. #5-A-%( X 28 ,#8 9'#9'-A#,8 :-*%-+ (,9. -,. 2.3{2.4). 32 0.4 0.3 fX (x) 0.2 0.1 0.0 ;8 N (;4U 1) N (0U 1) N (4U 1) .... ...... ...... ..... ......... ..... ....... ..... ....... ... ... ... ... ... ... ... ... ... .. ... ... . . . . . . ... . . ... ... .. ... ... ... .. ... ... . . . . . . ... . . . ... ... . . . . . . ... . . . ... ... . . . . . . ... . ... . ... . . . . . . . ... . . ... . ... . . . . . . ... . ... . ... . .. . . . . ... ... ... . . . . . . . . . ... ... . ... . . . . . . . . ... ... ... . . . . . . . . . ... ... . ... . . . . . . . . ... ... . ... . .. . . . . . ... .. ... . . . . . . . . . . ... ... ... . . .. . . ... . . ... .. ... .. . . ... . . . . ... .. ... .. . . ... . . . . ...... ..... . . .... . . . . . .... .. ....... ........ . . . . . . . . . . ...... ....... . .. ....... . . . . . . . . . . . . . . . . . . . . .. .. ... . .. .. .................................................................................................................................................................................................................................................................................................................................................................................................................. ;6 ;4 ;2 0 x 2 4 6 8 2.1 | $ fX (x) + 8 N (;4G 1), N (0G 1) N (4G 1) = 1 m = ;4 0 4 1.0 0.8 FX (x) 0.6 0.4 0.2 0.0 ;8 ............................................................................................................................................................................................................................................................. ............... ... ... ....... ....... ....... ...... . ..... ..... . . .. .... .... . . . . . . .... .... .... .... .... .... .. .. .. ... ... ... . . . .. .. ... ... ... ... .. .. ... .. .. . . . . . . .. .. .. .. .. ... .. .. .. ... ... ... . . . . . . .. .. ... ... ... ... ... .. ... ... ... ... . . . . . . .... .... .... .... .... .... .... .... ..... ....... ....... ....... . . . . . . . . . . . . . . . . . . . . . . . . .................................................................................................................................................................................................................................................. N (;4U 1) N (0U 1) ;6 ;4 ;2 N (4U 1) 0 2 x 4 6 8 2.2 | "+ FX (x) 8 N (;4G 1), N (0G 1) N (4G 1) = 1 m = ;4 0 4 0.8 0.6 fX (x) 0.4 0.2 0.0 ;8 ..... ... ... .. .... ... .... . ... .... ... ... .. ... ... ... . . ... . . ... .. . ... .. . . ... . . ... .. . .. . . . . .............. .... . . ..... .. ........ . ....... ..... . .... .. ...... .... ....... ........ . ... .... . . ... . . . ................................................. ..... . . ............ ................. . . . . . ... ............... ............ .. ... .... ............ .......... .. .. .... . ... ......... ....... ..... . . . . . . . ..... ...................... . . . .. . . ..... . . . . . . . ....... .............. . . . . ... . . . . . . . . ..... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....................................................................................................................................................................................... .............................................................................................................................................................. N (0U 0:5) ;6 ;4 ;2 0 x N (0U 1:0) N (0U 2:0) 2 4 6 8 2.3 | $ + fX (x) 8 N (0G 0 5), N (0G 1 0), N (0G 2 0) m = 0 = 0 5, 1,0 2,0 33 1.0 0.8 FX (x) 0.6 0.4 0.2 0.0 ;8 .......................... .......................................................................................................................................... ......... ................. ...... .......... ........ ............ ... ....................... ......... . . ...................... . .... . . . .... ... .. ..... ................. .. ....... ..... ... ... ............... .. .... ...... ........................... . . ........ .. .. .... ... .... ........... ..... .... ............................................ . ............. . ..... ........... ..... . . ..... ......... .......... .............. . . . . .... ... ... .... .. .. .... ... ... ..... .... .... . . . .. .. .... ..... ... ... ....... ... .. ....... ... ..... . . . . . . . . . . . . ... ..... ........ ......... ............... .................................................................................................................................................................................... N (0U 0 5) N (0U 1 0) N (0U 2 0) ;6 ;4 ;2 0 x 2 4 6 8 2.4 | "+ FX (x) 8 N (0G 0 5), N (0G 1 0), N (0G 2 0) m = 0 = 0 5, 1,0 2,0 & 3#*#5#%-+ (2.1) ,5#*2#', A' %95%# ,3#*#5#%-# 35%,'H 3#*#5+#',+ ,-9- *29+ 39#'9- | 9'#9'-A#,-9 :-*%-#9 m - *-,3#,-#. 2. %'#85%+ 42%$-+ ,3#*#5#%-+ FX (x) %95%. ,52A.%. #5-A-%( -9##' -* Zx (x0 ; m)2 ! 1 FX (x) = Pr(X < x) = p exp ; 2 2 dx0: (2:4) 2 ;1 39OH 42%$-- FX (x) 9:#' ?(' 3#*#5#% #+'%,' 33*%-+ %95%. ,52A.%. #5-A-%( &*%%(. -%'#5 ( ): Pr( X < ) = FX ( ) ; FX (): (2:5) '#-,'-A#,+ 42%$-+ q() %95%. ,52A.%. #5-A-%( X ,5#*2HO+ : 2 2 ! q() = MZexp(iX)] = exp im ; 2 : (2:6) *%-9 -& :%#.<-1 ,.,' %95%. ,52A.%. #5-A-%( +5+#',+ #E 2,'.A-,' 3- 93&-$--. -9#%%, 32,' X1 - X2 | %95%(# ,52A.%(# #5-A-%( , 39#'9- m1, m2 - 12, 22. 8* -1 ,299 X = X1 + X2 ?2*#' ':# %95%. ,52A.%. #5-A-%. , 39#'9- m = m1 + m2 - 2 = 12 + 22. 5+ 3-&5%(1 &%A#%-. 39#' m - '?25-' 42%$-- fX (x) FX (x) *,''A% ,5:%, 36'92 35&2H',+ ( ) (2:7) Z = X ; m *5+ '. 35'%,' ,3#*#5#%-+ fZ (z) ,5#*2HO+ : 2! 1 fZ (z) = p exp ; z2 (2:8) 2 '.#. #E 39#'( %( mZ = 0, Z = 1 -, '-9 ?&9, Z ! N (0U 1). 74- fZ (z), 3-#*#%%(. % -,. 2.1, %&(H' . 34 0.5 0.4 a(z) 0.3 0.2 0.1 ........................................................................................................................................................................... .................................. ............... ........... . . . . . . . . ... ....... ....... ..... ..... . . . ..... .... ... ..... . . .... .... ... ... . . .... ... ... . . ... ... .. ... . .. ... ... .. . . .. ... .. ... . . . ... ... ... . . ... .. 0.0 0 1 2 z 3 4 5 2.5 | J K L(z ) %'#85%+ 42%$-+ ,3#*#5#%-+ FZ (z) *5+ ,'%*'%. %95%. #5-A-%( 3#*#5+#',+ -& (:#%-+ Zz u2 ! 1 FZ (z) = Pr(Z < z) = p exp ; du: (2:9) 2 2 ;1 ,-52 ,-99#'-- ,?('-+ Z < 0 - Z > 0 %#+'%(, 36'92 FZ (0) = 0 5 Zz u2 ! 1 1 FZ (z) = 2 + p exp ; 2 du: 2 0 95%# ,3#*#5#%-# A,' -,35&2#',+ ,''-,'-A#,-1 (A-,5#%-+1. J2%$-+ ,3#*#5#%-+ FZ (z) &-,-' '5 ' *%. 3##9#%%. z. ,52 -%'#85 (2.9) %# (:#',+ A##& 65#9#%'%(# 42%$--, 3'-A#,-1 3-9#%#%-+1 35&2H',+ # 4 Zz u2 ! 1 exp ; 2 du a(z) = p (2:10) 2 0 - 36'92 FZ (z) = 0 5 + a(z): (2:11) J2%$-+ K35, +5+#',+ %#A#'%., a(;z) = ;a(z), 36'92 3- '?25-%-- 3-*+' '5 %#'-$'#5%(# &%A#%-+ 829#%' z. 5+ %95%. ,52A.%. #5-A-%( X 9:% &3-,' 3- x1 x2 x2 ; m x1 ; m ;a : (2:12) Pr(x1 X < x2) = a '9 ,52A#, 8* 8%-$( x1 - x2 ,358H',+ ,-99#'-A% '%,-'#5% m, '.#. x1 = m ; " - x2 = m + ", ' 9:% &3-,' ?5## 93'%. 49# " (2:13a) Pr(jX ; mj < ") = 2a 35 -5-, #,5- 3*,'-' " = z , Pr(jX ; mj < z ) = 2a(z): (2:13b) 58+ 3-#*#%%. 4925# z = 1, 2, 3, 4, -& '?5-$ 42%$-- K35, %.*#9 Pr(jX ; mj < ) = 2a(1) = 0 683U Pr(jX ; mj < 2 ) = 2a(2) = 0 954U (2:14) Pr(jX ; mj < 3 ) = 2a(3) = 0 9973U Pr(jX ; mj < 4 ) = 2a(4) = 0 99994: D' #%,' ?(A% *5+ z = 3 4925-2H' -*# % !: 3'-A#,- *,'#% ('.#. , #+'%,'H 0,9973), A' '5%#%-# %95% ,3#*#5#%%. ,52A.%. #5-A-%( ' #E 9'#9'-A#,8 :-*%-+ %# 3#(,-' 3 ?,5H'%. #5-A-%# '#1 ,-89. #+'%,' 3'-35:%8 ,?('-+ Pr(jX ; mj > 3 ) 3- 6'9 ,,'-' 0,0027. . 5 u1;=2 ,'%*'%8 %95%8 ,3#*#5#%-+ , 35'%,'H fZ (z), '#AHO-9 &*%%92 2%H &%A-9,'- , %&(#',+ '# &%A#%-#, 3- '9 (35%+#',+ #%,' Z u1;=2 Pr jZj < u1;=2 = ;u fZ (z) dz = ;=2 1 = FZ (u1;=2) ; FZ (;u1;=2) = 2a(u1;=2) = 1 ; : (2:15) 8#9#'-A#,. 'A- &#%-+ %1:*#%-# %'-5+ u1;=2 &5HA#',+ '9 (?# *21 8%-A%(1 &%A#%-. z, 3- '(1 35O*, 8%-A#%%+ ,#12 -. 35'%,'- fZ (z), ,H ?,$-,, ,%-&2 - #'-5%(9- 5-%-+9-, 31*+O-9- A##& 'A- z = ;u1;=2 - z = u1;=2, ?(5 ?( %. . 28-9,59-, *5+ %1:*#%-+ %'-5+ u1;=2 3- &*%%9 2%# %#?1*-9 #<-' 2%#%-# 2a(u1;=2) = 1 ; : (2:16) . %'-5 %&(#',+ . &9:% ':# -,35&%-# ! - ! %'-5#.. K#,'%%-. %'-5 ;u1; -O2' -& 2,5-+ Z ;u1; Pr (Z < ;u1;) = ;1 fZ (z) dz = 0 5 ; a(u1;) = (2:17a) ,'#','#%% 3,'%%-. %'-5 u1; | -& 2,5-+ Z1 Pr (Z > u1;) = fZ (z) dz = 0 5 + a(u1;) = 1 ; : u1; (2:17b) ,52A# %95%8 ,3#*#5#%-+ 5#,'%%-. - 3,'%%-. %'-5- %( 3 9*25H, 6' :# '%,-',+ ?-9 *2,'%%-9 %'-5+9. %#'%(# -1 &%A#%-+ 9:% %.'- % ,%%-- 4925 (2.16) - (2.17) -& '?5-$ 42%$-K35, (,9. 3-5:#%-#). *?% '#1%- 3-9#%#%-+ %95%. ,52A.%. #5-A-%( 3- #<#%-&*A 9'#9'-A#,. ,''-,'-- -&5:#% %-:#. 36 2.2. $ $ / *5+ ,-,'#9( -& *21 ,52A.%(1 #5-A-% (X Y) (%95%(. &% % 35,,'-) -9##' 35'%,' -* ( ) 1 1 q 2 exp ; f (x y) = (2:18) 2 ) Q(x y ) 2(1 ; rXY 2 X Y 1 ; rXY 2 (x ; mX )(y ; mY ) + (y ; mY )2 X) Q(x y) = (x ; m ; 2 r XY 2 X Y Y2 X 8*# mX mY | 9'#9'-A#,-# :-*%-+ ,52A.%(1 #5-A-% X, YU X Y | -1 ,#*%-# *'-A#,-# '5%#%-+U rXY | -1 644-$-#%' #5+$--. 5+ ,52A.%(1 #5-A-%, ,3#*#5#%%(1 3 %95%92 &%2, %##5-%%,' %,-5% %#&-,-9,'-. ,5- ,52A.%(# #5-A-%( Y %##5-%( (%#&-,-9(), ' rXY = 0 ( 2 2) 1 ( x ; m ) ( y ; m ) f (x y) = 2 exp ; 2 2 X ; 2 2 Y : (2:19) X Y X Y #+'%,' 33*%-+ ,52A.%. 'A-, ,3#*#5#%%. 3 %95%92 &%2, 3+9285%- R , ,'%9-, 355#5%(9- 85%(9 ,+9 ,,#-%-+, (:#',+ 4925. Prf( Y) 2 Rg = (2:20) ! !#" ! !# " a ; mY ; a ; mY (2:20) = a ; mX ; a ; mX X X Y Y &*#, x 2 Z ) y 2 Z ). + (655-3,9 ,,#-%-+) %&(#',+ 655-3,, ,#1 'A1 '8 ,9#,'%+ 35'%,' %95%8 &% 3,'+%% : f (x y) = const. 52,- 655-3, 33$-%5%( X Y : a = k X b = k Y . #+'%,' 33*%-+ ,52A.%. 'A-, ,3#*#5#%%. 3 %95%92 &%2, ?5,' k , 8%-A#%%2H 655-3,9 ,,#-%-+ , 352,+9- a - b, Prf(X Y) 2 k g = 1 ; exp(;k2=2) (2:21) 8*# k | &9#( 352,#. 655-3,, (:#%%(# ,#*%-1 *'-A%(1 '5%#%-+1: k = a= X = b= Y . ,5- X = Y = , ,,#-%-# 3 %95%92 &%2 %&(#',+ !. - 289 %95%9 ,,#-%-- , mX = mY = 0 ,,'+%-# R ' 'A- (X,Y) * %A5 *-%' ($#%' ,,#-%-+) ,3#*#5#% 6 f (r) = r= 2 exp ;r2=2 2 r 0: (2:22) ,3#*#5#%-H #5#+ 3*A-%+#',+ 9*25 #' % 35,,'-, #,5- #8 '8%5%(# ,,'5+HO-# (3#$-- % *-%'%(# ,-) %#&-,-9( - ,3#*#5#%( %95% , %25#(9- 9'#9'-A#,-9- :-*%-+9- - %(9- *-,3#,-+9-. 37 ??O#%-#9 &% #5#+ +5+#',+ 6 -6. - &*%%. %,'%'# s 35'%,' 6'8 &% ,5#*2HO+ : f (r) = r= 2 exp ;r2=2 2 I0 rs= 2 r 0 (2:23) 8*# Z2 ; 1 I0(x) = (2) expZx cos(')] d' 0 | # # * ! . 95%(. &% 3,'%,'# '#1 -&9##%-. *5+ %#&-,-9(1 ,52A.%(1 #5-A-% X, Y, Z f (x y z) = (2:24) ! 2 2 2 = (2 )3=2 1 exp ; (x ;2 m2 X ) ; (y ;2 m2 Y ) ; (z ;2 m2 Z ) : X Y Z X Y Z #+'%,' 33*%-+ ,52A.%. 'A- (X,Y,Z) ?5,' Ek , 8%-A#%%2H 655-3,-*9 %. 35'%,'- , 352,+9- a = k X , b = k Y , c = k Z , % q Prf(X Y Z) 2 Ek g = 2 a(k) ; 1 ; 2=k exp(;k2=2): (2:25) / *5+ ,-,'#9( -& n ,52A.%(1 #5-A-% X = (X1 X2 : : : Xn) (%95%(. &% #5-*9 3,'%,'# &9#%,'n) -9##' 35'%,' -* 1 1 ; 1 exp ; 2 (X ; Xc ) K (X ; Xc ) : (2:26) f (X ) = q n (2) det(K ) #' X ?&% -& n-93%#%'%8 %? - ,3#*#5#% 3 9%89#%92 %95%92 &%2 , #5+$-%%. 9'-$#. K - #'9 9'#9'-A#,-1 :-*%-. Xc = (Xc1 Xc2 : : : Xcn ). & (:#%-+ (2.26) ,5#*2#', A' 9'#9'-A#,# :-*%-# - *-,3#,-+ DZX] ,52A.%8 #' X ,5#*2HO-# : MZX] = Xc (2:27a) DZX] = MZ(X ; Xc ) (X ; Xc )] = Sp(K ): (2:27b) ', -,9+ *-,3#,-+ ,52A.%. #'%. #5-A-%( X % ,299# *-8%5%(1 65#9#%' Sp(K ) #5+$-%%. 9'-$( K . 2.3. { ! . . { I. 799-42%$-#. -5- -%'#859 D.5# %&(#',+ 42%$-+ -* ;() = Z1 0 x;1 exp(;x) dx: 38 (2:28) ?5,' 3#*#5#%-+ 899-42%$-- | ,+ A-,5+ ,, 9# %25+ '-$'#5%(1 $#5(1 A-,#5, = 0 ;1 ;2 : : :: 799-42%$-+ +5+#',+ -%'#859, &-,+O-9 ' 39#' . % 2*5#'+#' ,5#*2HO-9 ,.,'9: 1) ;( + 1) = ;() 3- > 0. (2.29) 2) ;(1) = ;(2) = 1. p 3) ; 21 = . (2.30) (2.31) -9 ?&9, #,5- | 35:-'#5%# $#5# A-,5, = n, ' ;(n) = (n ; 1)! -5;(n + 1) = n! (2:32) 5#*'#5%, 899-42%$-+ 9:#' ,,9'-',+ ??O#%-# 4'-5. ,5- :# 829#%' #% %25H -5- '-$'#5%92 $#592 A-,52, ' &%A#%-# 899-42%$-- ,1*-',+. 1 , ' ;() 9:#' ?(' 5#8 (A-,5#%. ,5- 829#%' 33$-%5#% 2 p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fT (tU ) = ;( ) t;1e;t t 0 -5- 42%$-#. ,3#*#5#%-+ Zt Pr(T t) = FT (tU ) = ;( ) u;1e;u du: (2:34) 0 ,3#*#5#%-# (2.33) -5- (2.34) %&(#',+ !- (-,. 2.7). % &-,-' ' ,#8 829#%' t - 39#' , . 9#%'( 3#8 - '8 3+* 3#*#5+H',+ -& #%,' MZT] = DZT] = 2 : (2:35) & (:#%-. (2.35) 9:% 352A-' $#%- 39#' 899-,3#*#5#%-+ 3 3('%(9 *%%(9 : = MZT] = DZT]U (2:36a) 39 10 8 6 4 2 ;() 0 ..... ..... ........ ... ... ... ... ... .... . ... . ... ... . ... ... .... .... .. . .... ..... ... . ..... .... .. ... ... ... .. .... ... ... ... .. ... .... ... . .... ..... ... ... ... ... ... ... ... ... ... . ... ..... ... ... ... ... ... ... ... ... . .. ;2 ;4 ;6 ;8 ;10 .. ... ... .. ... ... ... . ... .. ... ... ... ... ... .. . . ... . ... ... ... ... ... ... . ... . ... ... ... ... ... .. ... ... . ... .. ... ... ... ... ... .. . . ... .. ... .. ... ... ... .. ... . . . ... ... ... ... ... ... ... . . . ... ..... ... ..... ... ..... ..... ........ . . . ......... . . . . . .................................... . .. . . .. ..... . . ... . . .. ... . .... ..... ... ... . ... .. ..... ... .... ... ... ... .... . . . .... . . .. .. . ... ..... ... ... ... .... .. ... ... .. .. ... .... .. ... ... . ... . ... ... ... ... ... .. ...... ... .... ... .. ... ... ... . . ... . . . ... ..... ... ... . ... ..... ... . ... ... . ... .... ... .. ... ... ... . . ... . . . ... .... ... ... .... ... ... ... . ... ... . ... . . ... ... ... ... ... ..... ... . ... ... ... ... .... ... ... ... . . ... .. .. ... ... ... ... ... . . . ..... ....... ........ ...... ... ... .. .... .. .. . ... .. .. .. .. .. . ... .. . ... ... .... ... .. .. . . .... ... ... ... .. ... ... ... ... . . ... .... ... .. . ... ..... .. ... .... ... . ;4 ;3 ;2 ;1 0 1 2 3 4 5 2.6 | 8# +-# 7 ;() 1.00 0.75 fT (tU ) 0.50 0.25 ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. ... ... ... ... ... ... ... ... ... ... ... ... ... ......................................... . . . . . . . .......... .... .... . . .......... . .... ... . ......... . .... . .... ..... .... ............................................................................ . . . . . . . . . . . . . ......... ................ ........ . . . . . . . . . . . ......... .............. ..... . .... . . . . . ......... .............. . . . . . . ...... .... .......... ............. .. ....... ............... ...... .......... . . .. . . . . . . . . ........... .............. ......... .. . .. . . ................. . . . . . . . . . ............. ......... . .. ................. . . . . . . ................ . . . . ............ ..................... . .. . ................... . . . . . . . ....................... ............... . . . . . . . . . . .. ........................... . . . . . . . . . . ...................... . ........ ...................................... ......................................................................... . . . . . . . . . . . . . . . ................................................... .... ..... . .......... . . . . . . . . . . . . . . . . . . . . . ..................................................................................................................................................................................................................... ....... 0.00 0 =1 =2 1 =3 2 3 t 4 5 6 7 2.7 | $ fT (tG ) +- = 1,0 / / 40 0.5 0.4 f (2) 0.3 0.2 0.1 ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .......... ....... ....................... . . . ....... ... .. . . ...... ... .. .... .... . .................................................................. ... . . ...... ................................. . . . . ... ..... .. . ................................. . . . . .... ..................... .......... . . ... ....................... .............. .......... ............................ ... .... . . . ........ . ........... .............. . . . .......... .. . . . . . ......... .. ........... ............. . . . . . . ...... . .. ... . ............ ............... . . . . . . . . . . . . . . . . . . . . . ........... ......... .............. . .. . . .. . . . . . . . ................. ................................... . . . . . ........... ............... . .... ... . . . .................. ..................... . . . . . . . . . . . . . .................. ...................... . ..... .......... . . . . . . . . . . . . . . . . . . ........................................ ................................................................................................................................................................................................................................... ........ .............. . . . . . . . .............................................................................................................................................................................. . . . . . . . . . . . . . . . ... ............. 0.0 0 n=2 n=4 n=6 n=8 2 4 6 8 10 12 14 16 18 20 2 2.8 | $ f (2 ) 2 n = 2 4 6 8 = MZT]2 = DZT] = MZT]: (2:36b) - %#'(1 3#*#5#%%(1 &%A#%-+1 39#' 352A#9 A,'%(# ,52A899-,3#*#5#%-+. , 3- = 1 899-,3#*#5#%-# ,3*#' , 3&'#5%(9. - | 35:-'#5%9 $#59 A-,5# 352A#9 + !, <- -,35&2#9# '#-- 9,,8 ?,52:-%-+. '#-,'-A#,+ 42%$-+ 899-,3#*#5#%-+ ,5#*2HO+ : qT (z) = M Zexp(izT)] = (1 ; iz=); : (2:37) 2.4. / 2 (${) D' ,3#*#5#%-# ,+&% , %95%(9 - <- -,35&2#',+ 3- #<#%-&5-A%(1 &*A ,''-,'-A#,8 %5-&. ,%# ?&%-+ 6'8 ,3#*#5#%-+ 5#:-' ,,9'#%-# (?- -& %95%. ,23%,'-. ,,9'-9 ,52A.%2H #5-A-%2 Y, ,3#*#5#%%2H 3 %95%92 &%2 , 9'#9'-A#,-9 :-*%-#9 MZY] = a - ,#*%-9 *'-A#,-9 '5%#%-#9 , -5-, ?5## ', 32,' Y ! N (aU ). 8* ,52A.%+ #5-A-% U = (Y ; a)= , %&(#9+ , ,3#*#5#% 3 %95%92 &%2 , 39#'9- MZU] = 0 - U = 1, '.#. U ! N (0U 1). *' ,'%*'-&%%. ,52A.%. #5-A-%( !2 Y ; a 2 2 (2:38) U = %&(#',+ 2 (%-) $ . 41 ,,9'-9 '#3# n %#&-,-9(1 ,52A.%(1 #5-A-% Y1, Y2, : : : , Yn, ,3#*#5#%%(1 3 %95%92 &%2 , 9'#9'-A#,-9- :-*%-+9- a1, a2, : : : , an - ,#*%-9- *'-A%(9- '5%#%-+9- 1, 2, : : : , n. ?&2#9 *5+ :*. -& 6'-1 ,52A.%(1 #5-A-% $ %95%2H ,52A.%2H #5-A-%2 i = 1 2 : : : n: Ui = Yi ; ai i 299 *' ,'%*'-&%%(1 3##9#%%(1 2 = U21 + U22 : : : + U2n = (2:39) !2 !2 !2 Y Y Y 1 ; a1 2 ; a2 n ; an = + + ::: + 1 n 2 %&(#',+ , = n . ,''-,'-A#,-1 '?5-$1 - 3- (35%#%-- ,A#' A-,5 ,'#3#%#. ,?*( 3-%+' ?&%A' ?2. . 5'%,' ,3#*#5#%-+ ,52A.%. #5-A-%( 2 -9##' -* h i;1 f (t) = 2=2 ;(=2) t=2;1 exp (;t=2) (2:40) 3- 6'9 t 0. %8* 2*?% 35'%,' ,3#*#5#%-+ 2&(' %#3,#*,'#%% '#9-%1 2 : h i;1 f (2) d(2) = 2=2 ;(=2) (2)=2;1 exp ;2=2 d(2): (2:400 ) %-+ 35'%,' ,3#*#5#%-+ (2.40) , %## 3-#*#%%. 35'%,'H (2.33), &5HA#9, A' ,3#*#5#%-# 2 3#*,'5+#' ,?. A,'%(. ,52A. 899,3#*#5#%-+ 3- = n2 - = 21 . %'#85%+ 42%$-+ 2-,3#*#5#%-+ -9##' -* h i;1 Z 2 =2;1 F (2) = Pr(X 2) = 2=2 ;(=2) x exp (;x=2) dx: (2:41) 2 0 -9 ?&9, ,3#*#5#%-# 2 &-,-' ' *%8 39#' | A-,5 ,'#3#%#. ,?*(. -,. 2.8{2.9 -&?:#%( 3-9#( 84- 35'%,'- ,3#*#5#%-+ #+'%,'#. f (2) - -%'#85%. 42%$-- 2-,3#*#5#%-+ F (2) ,'#','#%%. 6' -*% -& 84- *5+ f (2), 35'%,' 2-,3#*#5#%-+ ,-99#'-A%, % -9##' 3(. 2*5-%#%%(. "1,'" ( 3#-4#-.%. ?5,'- ?5<-1 25%#%-. 2). ,'9 n (-5- ) ,-99#'-A%,' 35'%,'- 29#%<#',+, 3- 6'9 &% ,3#*#5#%-+ ,'#9-',+ %95%92. ,,A-' 3#(# * 99#%' 2, 352A-9 MZ2] = DZ2] = 2: (2:42) ,3#*#5#%-# 2 A,' -,35&2#',+ ,''-,'-A#,-1 (A-,5#%-+1, A,'%,'-, ,+&- , ,5#*2HO#. '##9.. 42 1.00 0.75 F (2) 0.50 0.25 ....................................................................................................................................................................................................................................................................................... ........................................ .................................................................................................................................................................................................................... ..................... .......... .......... ................ ............. . . . . . . . . . . . . . . ................. ................. . . . . . . . . . ....... .............. ................ ....... ........... ............ ............. . ........ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. ... ......... .......... ......... ..... ....... ......... ........ .... ...... ......... ........ .... ...... ........ ....... . . . ... . . . . . . . . . . . . . . . . . .. . .... ..... ..... ....... .... ..... ....... .... ..... ...... ..... ... ...... ..... .... . . ... . . . . . . . . . . . ..... ... ..... ..... .... ... ..... ... .... ..... .... ... .... ..... . ... . . .. . . . . . . . . . .. ..... .... ... ... ..... .... ... ... .... ..... .... ... ..... .... . . . . . . . . . . . .. .. ... ...... .... ... .... .... .. .... ..... ..... ... ..... .... .... . . . . . . . . . . . .. ... ..... .... .. .... ...... ... ... ..... ... ...... ..... ....... ... ....... . . . . . . . . . . . . . . . .. ..... ..... ... ..... .......... .............. ...................................................... 0.00 0 2 2 4 4 6 8 6 8 10 12 14 16 18 20 2 2.9 | "+ F (2 ) ( n = 2 4 6 8 # /) $ . , x1 x2 : : : xn | = 1 Pn xi ! N ( m ) ( n , & x n i=1 s2 = n;1 1 Pni=1 (xi ; x)2 | . ! X S2 | , (n ; 1) ;2S2 2n;1. '#-,'-A#,+ 42%$-+ ,3#*#5#%-+ 2 , A-,59 ,'#3#%#. ,?*( n = ,5#*2HO+ : h i q (z) = M exp(iz2 ) = (1 ; 2iz);=2 : (2:43) 2,' ,,9'-#',+ ,52A.%+ #5-A-% 2n , n ,'#3#%+9- ,?*( ,52A.%+ #5-A-% 2m , m ,'#3#%+9- ,?*(, ':# **-'-%+ ,52A.%+ #5-A-% 2n+m = 2n + 2m , n + m ,'#3#%+9- ,?*(. & -* 1'#-,'-A#,. 42%$-- (2.43) ('##', A' qn+m (z) = qn(z) qm(z) = (1 ; 2iz);(n+m)=2 (2:44) 35'%,' ,3#*#5#%-+ #+'%,'#. 93&-$-- 2n+m = 2n + 2m ,5#*2HO+ : n + m ;1 ( n + m ) = 2 fn+m (t) = 2 ; 2 t(n+m)=2;1 exp (;t=2) : (2:45) -9 ?&9, **-'-%+ ,52A.%+ #5-A-% 2n+m = 2n + 2m ':# 3-,(#',+ &%9 2 , n + m ,'#3#%+9- ,?*(, '.#. ,52A.%+ #5-A-% 2 ?5*#' ,.,'9 2,'.A-,'- 3- 93&-$--. 2,' -9##',+ n ,'%*'-&%%(1 3##9#%%(1 U21 U22 : : : U2n, % '(#, ,H A##*, %5:#%( s 5-%#.%(1 &-,-9,'#. (,+&#.). 8* ,3#*#5#%-# -1 ,299( *' ?2*#' 3*A-%+',+ &%2 (2.40), % , A-,59 ,'#3#%#. ,?*(, %(9 = n ; s. 3'-# ,''-,'-A#,-1 &*A AO# -,35&2H' %# ,92 35'%,' ,3#*#5#%-+ #+'%,'#. -5- 42%$-H ,3#*#5#%-+, 2- , '#AHO-# $ , '(# 3- &*%%9 ?&%AH' 2 . 43 5 2 , '#AHO-9 &*%%92 2%H &%A-9,'- , %&(#',+ '# &%A#%-# 2 = 2 , 3- '9 (35%+#',+ #%,' Z1 Pr 2 > 2 = 1 ; F (2 ) = 2 f (2) d(2) = : (2:46) . 8#9#'-A#,. 'A- &#%-+ %1:*#%-# %'-5+ 2 &5HA#',+ '9 (?# &%A#%-+ 2 = 2 , 3- '9 35O*, 8%-A#%%+ ,#12 -. 35'%,'- f (2), ,H ?,$-,, ,%-&2 - #'-5%. 5-%-#., 31*+O#. A##& 'A2 2 = 2 , ?(5 ?( %. . 28-9- ,59-, *5+ %1:*#%-+ %'-5+ 2 3- &*%%(1 - %#?1*-9 #<-' 2%#%-# (2.46). 0.10 f (2) 0.05 5O* = ......... . ............. ...................... ... ........ ........ ... ...... .... ....... ... .... . . . . . . ...... . ... ....... .... ... ...... ... .... ...... ... .... . . . . . . . . . ....... .. ... ....... ... .. ......... ... .... ... .......... ... ... . . . . . . . . . . . . . . . . . .... .... ..... ............... ..... ... ... ... ... ... ................ ... ... ... .... .... ... ... .................. ... . ... ... ... ... ... ... ... ............... . .. . . .. ... ... .. ... .. ... ... ......................... . . . ... ... ... ... ... ... ... ... ... .. ... .................... . . .. . ... ... ... ... ... ... ... ... ... ... ... ... ... ..................................... . . . .. . . . . . . .... ... ... ... ... ... ... ... ... ... .. ... ... ... ... ... .. ............................................................................................. . . .................. . ... ... .. 0.00 0 1.0 0.8 F (2) 0.6 0.4 0.2 0.0 0 5 10 15 210 20 25 2 ........................................... ........................................................................................ ................................. ...................... . . . . . . . . . . . . . . ...... ................ ................ ................ ................ ................ ................. ................................................... ................ ........ . ......... ... . . . . . . . . .... . . . . . . .. ... ........ ....... ... ....... ...... . . . . . . .... ... ....... . . . . . . ... ... ....... . . . . . . . ..... .. ........ . . . . . . . . ..... ... .......... . . . . . . . . . . . . . ................................................ ... .... F (210) = 1 ; 5 10 15 210 20 25 2 2.10 | $ f (2 ) (/) + F (2 ) () = 10 ( = 0 20G 2 = 202 10 13,20) -,. 2.10 (,#12), % '9 3-#*#% 35'%,' f (2) *5+ ,52A+ = 0 20 - = 10, 6' 35O*, A-,5#%% %+ 2%H &%A-9,'- , &<'-1% #'-5%(9- 5-%-+9-. '9 :# -,2%# (,%-&2) 3-#*#% -%'#85%+ 42%$-+ ,3#*#5#%-+ F (2), ,,9'#% '' :# ,52A. ( = 0 20 = 10). 2%'-9 % -,2%# '9#A#% 2#% &%A-9,'- - '#AHO-. #92 %'-5 2 . 3-5:#%-- 3-#*#%( &%A#%-+ %'-5#. 2 *5+ &5-A%(1 &%A#%-. A-,5 ,'#3#%#. ,?*( - 2%+ &%A-9,'- . 44 0.4 0.3 fT (t) 0.2 0.1 N (0 1) =3 =1 . .................. .. .... ... ... . ... .. ... .......................... ..... . . . . . . . .. ..... .......... ..... ... . ...... .. ... ...... ... ... ..... ... ... ...... ...... ... ... ..... ... ......... . . . ...... ... . . .. .. . ... ... ... ... .. . .. .. ... ... ... ... .. .. ... ... ... .. ... ... ... ... ... .... ... ......... . . . ... ... ... .. ... ... .. . .. ... ... ... ... ... ... .. ... .. ... .. ..... ... ... ............. . ... .... .... .. ... . . ... ... . ... ... ...... .. . . ... ... . ... ....... ... . . ... ... . ... .......... . . .. .. . ... ......... . ...... ... .......... . ... ..... ..... .... . ... ...... ... .. . ... ..... .... ... . . .......... ....... . . ........ ....... . . .......... ........ . . ........ ....... . . . ......... .......... ....... . . . . . ....... . . .. . . ........ . . . . ..... ......... . . . . . .......... ................ . . . . . . . . ... .............. ............. . . . . . . .... ............... . ........... .... . . . . . . . . ..... ........................ . ............... ..... . . . . . . . . . .... ........... ................... . . . . ...... ............... ....... . . . . . . . ... ...... . ....... . . . . . . . . . . . . . . . .. ........ .......................................................................................... ............................................................................................... ................... ................ ... ............................................................................................................ ........................................................................................... 0.0 ;5 ;4 ;3 ;2 ;1 0 t 1 2 3 4 5 2.11 | $ '1 fT (t) / = 1, = 3 = 1 | + N (0G 1) 2.5. / 2 ,3#*#5#%-# 'H*#%' (t-,3#*#5#%-#) -9##' :%# &%A#%-# 3- ,''-,'-A#,-1 (A-,5#%-+1, ,+&%%(1 , %95%(9 &%9, -9#%% '8*, 8* ,#*%## *'-A#,# '5%#%-# %#-&#,'% - #O# 3*5#:-' 3#*#5#%-H 3 3('%(9 *%%(9. 2,' Y - Y1 Y2 : : : Yn | %#&-,-9(# ,52A.%(# #5-A-%(, -9#HO-# %95%# ,3#*#5#%-# , 39#'9MZY] = MZY1] = MZY2] = : : : = MZYn] = 0 - ZY] = ZY1] = ZY2] = : : : = ZYn] = 1: 52A.%+ #5-A-% !;1=2 X ;1=2 n 1 2 = Y n1 2n (2:47) T = Y n Yi i=1 +5+HO+,+ 42%$-#. %95% ,3#*#5#%%(1 ,52A.%(1 #5-A-%, %&(#',+ $ '$ . 2,# '#-- #+'%,'#. 3&(#',+, A' 35'%,' ,3#*#5#%-+ #+'%,'#. ,52A.%. #5-A-%( T -9##' -* (,9. 3-9#( &*#52) ;(+1)=2 ; +1 ;1 < t < 1: (2:48) fT (t) = S (t ) = p 2 1 + t2= ; 2 D' ,3#*#5#%-# %#39#'-A#,#, '.#. % %# &-,-' ' 39#' -,1*%(1 ,52A.%(1 #5-A-%, &-,-' 5-< ' -1 A-,5. 45 0.4 0.3 fT (t) 0.2 0.1 0.0 .... ....... .......... ... .... ... .. . . ... ... ... . ... . .. . ... . . ... . . . ... . . . ... . ... ... . . ... . . ... . .. ... . . ... . . . ... . .. ... . . . ... . . . ... . .. ... . . . .... . . . . .... . . . .... .. . . . .... .. . . . .... . . . ..... . . .. . ....... . . . . .. . ......... . . . . . . ... .... . ... .............. . . . . . . . . . . ... ... ....................... ........ ... ... . . . . . . . . . . . . . . . . . ......... ... ... ... .. .. . . . . . ... ... ... .... ... ................................................................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........................ . .. .. .. .. .. .. .. .. .. .... ... . . . . . . .. .. .. . . . ......... ... ... ... ... . . 5O* = = =2 5O* = = =2 ;5 ;4 ;3 ;2 ;1 ... ... ... .. ;t=2 0 t 1 2 3 4 5 ... ... ... .. t=2 2.12 | $ fT (t) '1 = 3 (0/ 2 ) 6'. 4925# ?2. ?&%A#% A-,5 ,58#9(1 3*#%%9 (:#%-*?- 'H*#%', '.#. = n. # ?&%A#%-# A-,5 ,'#3#%#. ,?*( ?O#3-%+' 9'#9'-A#,. ,''-,'-#, ' ?5#8A#' 35&%-# ,''-,'-A#,-9'?5-$9-. & 4925( (2.48) ,5#*2#', A' ,3#*#5#%-# ,52A.%. #5-A-%( T %# &-,-' ' 39#' ,3#*#5#%-+ %#&-,-9(1 ,52A.%(1 #5-A-% Y - Y1 Y2 : : : Yn, &-,-' '5 ' *%8 39#' |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t-,3#*#5#%-+ ,358#',+ (<# %95%. -.). 9'#9'-A#,. ,''-,'-# *,''A% A,' -,35&2H',+ %'-5- t=2 ,3#*#5#%-+ 'H*#%' &-,-9,'- ' A-,5 ,'#3#%#. ,?*( - &*%%8 2%+ #+'%,'- . %A#%-+ %'-5#. ,3#*#5#%-+ 'H*#%' t=2 982' ?(' %.*#%( -& #<#%-+ 2%#%-+ Z1 Pr jTj > t=2 = 2 fT (t0) dt0 = : (2:50) t=2 46 1.0 0.8 fF (x) 0.6 0.4 0.2 ........ ... ... .. .... .. ... . . ... ... ... ... ... . ... . . ... . .... . ........... ... ... . . ... .... .... ... . . . . . ... . .. ... ... ..... ... .... ... ... ........ ..... .... ... .... ... ... ............. . . . . . . ... ... ... ... ....... . ... ... .... .... ... ....... ... ... .... ... ......... .... ... ... ... . . . . .... ........ ..... . . .. .... .. ... .. .... . ... .. .... .... ... ...... ... ... ... ... ... ... ...... ..... ... ... .... ... . . . ... ........ ... . . . . . . ... ... . .. ... .. .... .... ... ... ... .... .. ... ...... .... ...... ... .... .. .... ..... ...... ... ... ... . .... .......... ...... .... . .... . . ... . ........ . . . .... ... ...... . . . .... .... .. .. .... .... ..... .............. .. .... ........ .... ......... ...... ... ....... ...... .... .. ... .......... . . . . . . . . . ....... ... .... ... ........ .. .... .. ... ...... ... ........... ...... .... ........ ... ................... ... .... .................. ... .......... . . . . ... .... ... ...... ..... .............................................................. ....... ...... ... ... ...... ......... ................................ ....................................................................................................... .... .......... .... .............. .............. . . . . ......................................................................................................................................................................................................................................................................................................................... ........... ...................................................................................... ..................................................................... 0.0 0 1 = 10 2 = 1 1 = 10 2 = 10 1 = 10 2 = 100 1 2 3 x 4 5 6 7 2.13 | $ J0 fF (x) 1 2 8#9#'-A#,. 'A- &#%-+ %1:*#%-# %'-5#. ,3#*#5#%-+ 'H*#%' t=2 &5HA#',+ '9 (?# &%A#%-+ t= 2 , 3- '9 ,299%+ 35O* 3* -. 35'%,'- fT (t) % 2A,'1 ;1 ; t=2 - t=2 1 ?(5 ?( %. . -,. 2.12 ,299%+ 35O* &<'-1%%(1 *21 2A,' ,,'5+#' . 2.6. / 3* 6 93 (F -,3#*#5#%-#) 23'#?5+#',+ 3- ,%#%-- *-,3#,-. %95%(1 ,3#*#5#%-., (A-,5#%%(1 % ,%%-- 3('%(1 *%%(1. (D'' &% #O# A,' %&(H' ,3#*#5#%-#9 J-<#-%#*#). 2,' ,52A.%(# #5-A-%( X1, X2, : : :, Xm - Y1, Y2, : : :, Yn %#&-,-9( - -9#H' %95%# ,3#*#5#%-# , 39#'9MZXi ] = 0U DZXi] = 1U i = 1 2 : : : mU MZYj ] = 0U DZYj ] = 1U j = 1 2 : : : n: #&&9#%+ ,52A.%+ #5-A-% 1;1 X !0 X m n 1 1 F = m X2i @ n Yj2A (2:51) i=1 j =1 ,3#*#5#% 3 &%2 ,3#*#5#%-+ J-<#, '.#. -9##' 35'%,' ,3#*#5#%-+ #+'%,'#. (x 0) 1 + 2 1=2 ; (1 +2 )=2 ; x;1+1 =2 1 + 1 x (2:52) fF (x) = 1 2 2 1 2 2 ; 2 ; 2 47 1.5 f () 1.0 0.5 0.00.0 ......... .... ....... ... .... ... ... . . . ... ... ... . . ... . . . ... . . ... . . . ... . . . ... . . ... . . . ... . . ... .. . ... . . . ... . . . ... .. ..................................................................................................... . ... . .............................................. . . ... ................... .. . .. ................... . . . . . . . ... .. .......... .. ...... ... ... ..... ..... ... .... . . ... . . . ... ... .. ... .... ... ... .... . ... . . . ... . .. . . .... . . . . . . . ... . . . .. .... . . . . ... .. .. . . . .... . . . . . . ..... .. . . . . . . .... . .. . . . ..... . . . . ...... . . . . .. . ........ . . . . . .......... .. . . . . . .............. . . . . . . . . . .......................... . . . ................................................................................ ................................................. f () 0.5 1.0 K () 1.5 2.0 1.0 K () 0.5 0.0 2.14 | $ f () + F () , :+ 8*# 1 = m | A-,5 ,'#3#%#. ,?*( A-,5-'#5+U 2 = n | A-,5 ,'#3#%#. ,?*( &%9#%'#5+ (2.51). & 4925( (2.52) ,5#*2#', A' ,3#*#5#%-# ,52A.%. #5-A-%( F &-,-' ' *21 39#' | 1 = m - 2 = n. 5+ 2*?,' 35&'#5#. ,''-,'-A#,-9- '?5-$9- A-,5 ,'#3#%#. ,?*( 3-3#:%#92 ?&%AH',+ ?29- 1 = m - 2 = n. 74- 35'%,'- #+'%,'#. F ,3#*#5#%-+ -&?:#% % -,. 2.13. 9'#9'-A#,. ,''-,'-# *,''A% A,' -,35&2H',+ %'-5- F ,3#*#5#%-+ f 1 2 &-,-9,'- ' A-,5 ,'#3#%#. ,?*( 1 - 2 - &*%%8 2%+ #+'%,'- . & #<#%-+ 2%#%-+ Z1 Pr ( jFj > f 1 2 ) = 2 fF (x) dx = (2:53) f 1 2 982' ?(' %.*#%( &%A#%-+ %'-5#. f 1 2 ,3#*#5#%-+ J-<#. 2.7. / 4& 3'-A#,-1 &*A1 3#- 8-3'#& ,85,-- *%%(1 (?- , %#'%(9 '##'-A#,-9 &%9 ,3#*#5#%-+ *5+ 5H?. %#3#(%. ,52A.%. #5-A-%( 3-9#%+#',+ --'#-. .. 598. &%-HO+ 3- 3## '. 8-3'#&( ,52A.%+ #5-A-% ((?A%+ ,''-,'-) y -9##' -%'#85%2H 42%$-H ,3#*#5#%-+ K () -* (-,. 2.14) 1 X (;1)k exp(;2k22) (2:54) K () = Pr (y < ) = 1 ; k=;1 48 1.0 f () 0.5 f () K () ........................ ...... .... .... .... ... .... . . ... . . . .... . . ... . .. ... ..................................................................................................................................... . . ... .................... . . ... . ............. . . ... .......... . . . . .. . . . . . ... ....... .. ... ...... ... ... ... ....... .. ..... ...... . . . . . . ... . ... . . . . . ... .. ... ... .. .... ... ..... ... . ... . . . . . . ... . .. . . . ... .. . . . . . . .... . .. . . . . ... . . .. . . .... . . . . .. . . .... . . . . . .... . . . . . .... .. .. . . . . .... . .. . . . ..... . . . . .. . ..... . . . . . . . ..... . . . . .. ....... . . . . . . . ........ . .. . . . . . .......... . .. . .. . .............. . . . . . ........... ....................... . . ................................................................................ ............... 0.00.0 0.5 1.0 1.5 2.0 1.0 0.5 K () 0.0 2.15 | $ f () + K () , :+-' - 35'%,' ,3#*#5#%-+ #+'%,'#. 1 d K () = 4 X (;1)k k2 exp(;2k22): f () = d k=;1 (2:55) :#, - ,3#*#5#%-# 'H*#%', 6' ,3#*#5#%-# %#39#'-A#,#, '.#. % %# &-,-' ' 8-5-? 39#'. *%,'%%-1 -'#-+1 ,85,-+ (?A%. ,''-,'-- y , &*%%(9 '##'-A#,-9 ,3#*#5#%-#9 ':# -,35&2#',+ ,3#*#5#%-# 5989-%, 2 '8 -%'#85%+ 42%$-+ ,3#*#5#%-+ K () ,5#*2HO+ (-,. 2.15) K () = Pr (y < ) = 1 ; exp(;22 ) (2:56) , '#AHO#. #. 35'%,'H ,3#*#5#%-+ #+'%,'#. d K () = 2 exp(;22): f () = d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r(T < 30) = FT (30 ). a) *+, (2.34) = 640, = 20 - t = 30, 352A-9 640 Z30 Pr(T < 30) = FT (30 640 20) = ;(640) x639e;20x dx = 0 057: 0 ', *#4-$-' ?2*#' -9#' 9#,' , #+'%,'H 0,057. ?) &*%%. #5-A-%# p = Pr (T < 30) = 0,01 = FT (30 20) %1*-9 = 660. - '9 ?T#9# 3,' #+'%,' *#4-$-' %# 3#(,-' 0,01. '#','#%%, , #+'%,'H 0,99 *#4-$-' & ,,9'-#9(. #9#%%. -%'#5 %# ?2*#'. ' 2.3 .'- 9'#9'-A#,-# :-*%-+, *-,3#,-- - 644-$-#%' #5+$-- *29#%. %95%. ,52A.%. #5-A-%(. () 3-<#9 (:#%-# *5+ 35'%,'- ,3#*#5#%-+ fXY (x y) %95%. ,-,'#9( -& *21 ,52A.%(1 #5-A-% (X Y) ) ( 1 1 q 2 exp ; () f (x y) = 2 ) Q(x y ) 2(1 ; rXY 2 X Y 1 ; rXY 2 (x ; mX )(y ; mY ) + (y ; mY )2 X) Q(x y) = (x ; m ; 2 r XY 2 X Y Y2 X 8*# mX , mY , X , Y , rXY | 39#'( &%. 1) A#-*%, A' Z1 Z1 Z1 MXY ZX] = x fXY (x y) dx dy = x fX (x) dx = MX ZX] ;1 ;1 ;1 8*# fX (x) | 3$-5%+ 35'%,' X-93%#%'(: ( 2) ( x ; m 1 1 X) fX (x) = p exp ; 2 2 2 X X 352AHO+,+ -& (:#%-+ () -%'#8-%-#9 3 3##9#%%. y. 50 %'#8-2+, 352A-9 *5+ 3#(1 *21 99#%' X-93%#%'( MXY ZX] = mX DXY ZX] = DX ZX] = X2 : 2) %58-A% 3,'23+ *5+ Y-93%#%'(, %.*#9 MXY ZY] = mY DXY ZY] = DY ZY] = Y2 : 3) 3-<#9 (:#%-# *5+ ,9#<%%8 '8 99#%' Z1 Z1 kXY = MXY Z(X ; mX )(Y ; mY )] = (x ; mX )(y ; mY ) fXY (x y) dx dy: ;1 ;1 &%-<#9 *.%9 -%'#85# 3##.*#9 %(9 3##9#%%(9 -%'#8-%-+ (u v) 3 3-52: u = (x ; mX )= X , v = (y ; mY )= Y , A' *#' u2 ; 2r uv + v2 Z1 Z1 1 q kXY = uv exp ; 2(1 ;XYr2 ) du dv: 2 2 1 ; rXY ;1 ;1 XY #3# 2*?% 9#,' 3##9#%%. u #,'- #OE *%2 3##9#%%2H -%'#8-%-+ 3 3-52 t = u ; rXY v: t2 + (1 ; r2 )v2 Z1 Z1 1 (t + rXY v)v exp ; 2(1 ; r2XY) dtdv: kXY = q 2 2 1 ; rXY XY ;1 ;1 6'9 -%'#85# 5* ' 3#8 ,58#98 285(1 ,?1 (t + rXY v) ':*#,'#%% #% %25H -&-& %#A#'%,'- 3*(%'#85%8 (:#%-+ - A#'%,'3#*#5 -%'#8-%-+ '%,-'#5% 3##9#%%. t. ,'<-.,+ *.%. -%'#85 ,'%-',+ 3-&#*#%-#9 *21 *%'%(1 -%'#85: r Z1 2 Z1 2 1 t XY 2 exp ; v dv: p kXY = q exp ; dt v 2 ) 2 ) 2(1 ; rXY 2 2 ;1 2(1 ; rXY ;1 2A#'9 %9-A%8 9%:-'#5+ 3#(. -& 6'-1 -%'#85 ':*#,'#%% #% 1. '. -%'#85 (?#& ,9%:-'#5+ rXY ) ,3*#' , &%A#%-#9 *-,3#,-3##9#%%. v, '+ ':# % 1. ', kXY = rXY : -9 ?&9, &% *29#%8 ,3#*#5#%-+ () 35%,'H 3#*#5+#',+ &*%-#9 #8 A-,5(1 1'#-,'- mX , mY , X , Y - rXY . 51 ' 2.3 (( 2 n 2) *% %? %#&-,-9(1 %95%(1 #5-A-% fX1 X2 : : : Xng , 39#'9fm1 = 0 m2 = 0 : : : mn = 0g - f 1 = 1 2 = 1 : : : n = 1g. ,,9'-9 ,52A.%2H #5-A-%2 Y = n X i=1 X2i : 52A-' &% ,3#*#5#%-+ ,52A.%. #5-A-%( Y. () 2,' ,%A5 n = 1. 6'9 ,52A# 1'#-,'-A#,+ 42%$-+ qY1 () % Z1 1 2 qY1 () = MZ exp(iY1)] = MZ exp(iX1)] = p exp ;x21=2 + ix21 dx1: ;1 2 %'#8-2+, %.*#9 qY1 () = (1 ; 2i);1=2: ,52 3 2,5-H 93%#%'( Y %#&-,-9(, ' ,52A#, 8* % ?&% -& n ,58#9(1, 352A-9 n h in Y qY () = qYi () = qY1 () = (1 ; 2i);n=2 : i=1 5'%,' ,3#*#5#%-+ fY (t) Y 9:% 3#*#5-' , 39OH ?'%8 3#?&%-+ J2#, A' *#' fY (t) = Ct n2 ;1e;t=2 t 0 8*# C | 3,'+%%+. E 3#*#5-9 -& 2,5-+ %9-Z1 fY (t) dt = 1 0 '2* C ;1 = 2n=2 ;(n=2). ', fY (t) = 2n=2 ;(1 n=2) tn=2;1e;t=2: .*#%%+ 35'%,' '#A#' ,52A.%. #5-A-%#, ,3#*#5#%%. 3 &%2 2 , n ,'#3#%+9- ,?*(. ' 2.4 (( 45) *% %? %#&-,-9(1 %95%(1 #5-A-% fX1 X2 : : : Xng , 39#'9fm1 = 0 m2 = 0 : : : mn = 0g - f 1 = 1 2 = 1 : : : n = 1g. 9# '8, -9##',+ #O# *% %#&-,-9+ %95%+ #5-A-% X0 , 39#'9- m0 = 0 - 0 = 1. ,,9'-9 ,52A.%2H #5-A-%2 !;1=2 X n 1 2 T = X0 n Xi : i=1 52 52A-' &% ,3#*#5#%-+ ,52A.%. #5-A-%( T. () ,,9'-#9+ ,52A.%+ #5-A-% T +5+#',+ ,'%*'-&%%(9 '%<#%-#9 'H*#%' T = q X20 : =n 5+ %1:*#%-+ 35'%,'- ,3#*#5#%-+ fT (t) ,35&2#9,+ '#1%-. 42%$-- - 2 0 13 X fT (t) = M 4 @ q 20 ; tA5 : =n *#, 9'#9'-A#,# :-*%-# %1*+' '%,-'#5% 2 - X0 , 35'%,'+9- ,3#*#5#%-. fX0 (x0) dx0 = 21 exp ;x20=2 dx0 f 2 (2) d2 = 2n=2 ;(1 n=2) (2) n2 ;1e; 2 =22 d2: *,'-9 6'- 35'%,'- (:#%-# *5+ fT (t) - 2A'#9 4-5'2HO## ,.,' -42%$--, '8* 352A-9 Z1 22 2! 1 1 t 1 2 n ; 1 fT (t) = 2n=2 ;(n=2) n ( ) exp ; 2 n ; 2 d2: 0 ,35&2+ '#3# %2H 3##9#%%2H -%'#8-%-+ u 3 3-52 1 t22 + 2 = u 2 n 2 352A-9 " p 2 !#;1 Z1 fT (t) = 2n=2 n 1 +2t =n u(n;1)=2e;u du: 0 5&2+, 3#*#5#%-#9 ;-42%$--, %A'#5% %.*#9 n+1 ) 2 !;(n+1)=2 ;( t 2 fT (t) = pn ;( : n) 1 + n 2 .*#%%+ 35'%,' '#A#' ,52A.%. #5-A-%#, ,3#*#5#%%. 3 &%2 'H*#%'. ' 2.5 (( 6){) *%( *# ,52A.%(# #5-A-%( S - T, 3*A-%+HO-#,+ &%2 2 ,'#','#%% , n1 - n2 ,'#3#%+9- ,?*( - -9#HO-# *-% - '' :# 39#' . 52A-' &% ,3#*#5#%-+ ,52A.%. #5-A-%( U = S=T. 53 () % ,3#*#5#%-+ -,1*%(1 #5-A-% ,5#*2HO-. (&*#, C1 - C2 | %9-A%(# %,'%'() : s n1 ;1 1 s2 ! ds exp ; 2 2 s 0 fS (s) ds = C1 t n2 ;1 1 t2 ! dt fT (t) dt = C2 exp ; 2 2 t 0: R'?( 3#*#5-' 492 &% #5-A-%( U, ,,9'-9 ,3#*#5#%-# 3( S - T. #*#9 ,52A.%(# #5-A-%( R, U - ,*#5#9 &9#%2 3##9#%%(1 t = u s2 + t2 = r2 u r 0 s -5s = p r 2 t = p ru 2 : 1+u 1+u ',H* D(s t) = p r : D(r u) 1 + u2 8* 65#9#%' *-44##%$-5%. #+'%,'- fRU (r u) dr du 3( (R U) &3-<#',+ -*# (C3 | %9-A%+ %,'%') 2 ! n1 +n2 ;1 dr un2 ;1 du : fRU (r u) dr du = C3 exp ; 2r 2 r (1 + u2)(n1+n2 )=2 ',H* ,5#*2#', A' #5-A-%( R - U | %#&-,-9(, 3- 6'9 #5-A-% U 3*A-%+#',+ &%2 2 , n1 + n2 ,'#3#%+9- ,?*(. -%'#8- 3,5#*%## (:#%-# 3 r, 352A-9 65#9#%' *-44##%$-5%. #+'%,'- U 49# un2;1 fU (u) du = C4 du (1 + u2)(n1+n2 )=2 8*# C | %9-A%+ %,'%'. D'2 3,'+%%2H %.*#9 -& 2,5-+ %9-1 R f (u4 ) du = 1. ', 35'%,' ,3#*#5#%-+ U ,5#*2HO+ : U 0 n2 ;1 2 ; n1+2 n2 u fU (u) = n1 : ; 2 ; n22 (1 + u2)(n1+n2 )=2 .*#%%+ 35'%,' '#A#' ,52A.%. #5-A-%#, ,3#*#5#%%. 3 &%2 J-<#-%#*#. ' 2.6 5+ ,52A.%. #5-A-%( X, 3*A-%+HO#.,+ : ) ?-%9-5%92 ,3#*#5#%-H, ?) ,3#*#5#%-H 2,,%, %.'- #E 1'#-,'-A#,2H 42%$-H q(). 54 () ,35&2#9,+ 3#*#5#%-#9 1'#-,'-A#,. 42%$-X q() = MZ exp(iX)] = Pi exp(ixi): i 5+ ?-%9-5%8 ,3#*#5#%-+ , 39#'9- p - n -9##9 n q() = pei + 1 ; p : 5+ ,3#*#5#%-+ 2,,% , 39#'9 a %.*#9 h i q() = exp a ei ; 1 : ' 2.7 &*A1 $#%-%-+ 1'#-,'- 3 8%-A#%%92 A-,52 %?5H*#%-. n, -*#%'-4-$-- 3$#,, - *. 3#*,'5+#',+ $#5#,?&%(9 -,35&%-# ,''-,'-n) ; X(n) R = X(max min n) - X(n) | .%-# A5#%( %&(#9. &919 -5- <-'. (?- (X(max min -$-%%8 +*). ?O#9 -*# ,3#*#5#%-# &91 3#*#5+#',+ ,5#*2HO-9 ?&9 : Z1 (n) ( n ) Pr (Xmax ; Xmin) < t = n ZFX (x + t) ; FX (x)]n;1 dFX (x): ;1 ,35&2+ 6' #&25'', 3-#*#9 (?#& *&'#5,') %#,5 2'#:*#%-., '(# &(H',+ 35#&%(9- '#-- $#%-%-+. 8 1. 5+ (?- -& %9#% ,3#*#5#%%. 8#%#5%. ,23%,'- , 35'%,'H ,3#*#5#%-+ fX (x) = b ;1 a x 2 (a b) 35'%,' ,3#*#5#%-+ &91 R -9##' -* n;2 t fR (t) = n(n ; 1) (b ; a)n;1 : 8 2. 2,' Y | $#%'-%%+ - %9#% ,3#*#5#%%+ ,52A.%+ #5-A-% Y 2 (;a a). 8* 35'%,' ,3#*#5#%-+ #E &91 R -9##' -* fR (t) = 21a : 8 3. 2,' ,52A.%+ #5-A-% X -9##' %95%# ,3#*#5#%-# , %25#(9 9'#9'-A#,-9 :-*%-#9 - *-,3#,-#. 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P = 1 ; , '8* 352A-9 uZ=2 ! x ; a 2 Pr ;u=2 < =pn < u=2 = p exp ;t2=2 dt = 1 ; : 2 0 68 (3:30) #<+ 2%#%-# uZ=2 2 p exp ;t2=2 dt = 2 a u=2 = 1 ; 2 0 (3:31) %1*+' ,'%*'-&%%8 %95%8 ,3#*#5#%-+ u=2. ?(A% %'-5- %95%8 ,3#*#5#%-+ u=2 %1*+' 3 '?5-$9 K35, (,9. 3-5:#%-#) -& 2,5-+ 2 a(u=2) = 1 ; . 5+ %-?5## 23'#?-'#5%(1 &%A#%-. *#-'#5%. #+'%,'- P = 1 ; %'-5- ,'%*'-&%%8 %95%8 ,3#*#5#%-+ 3-#*#%( '?5. 3.1. ) 3.1 | : + + #-'#5%+ #+'%,' P = 1 ; 0,90 0,95 0,99 0,9973 0,999 %'-5 u=2 1,64 1,96 2,58 3,00 3,37 2*#9 ,A-'' %'-5 u=2, ,'#','2HO-. &*%%. *#-'#5%. #+'%,'- P = 1 ; , -&#,'%(9. #?&2#9 %##%,', ,'+O## 5#. A,'2%#%-+ (3.29), ! Pr ; pn u=2 < (x ; a) < pn u=2 = 1 ; -5! (3:32) Pr x ; pn u=2 < a < x + pn u=2 = 1 ; : p p 5#*'#5%, *#-'#5%(. -%'#5 Zx; u=2= nU x+ u=2= n] %(#' %#-&#,'%# 9'#9'-A#,# :-*%-# a , &*%%. #+'%,'H 1;. pA%,' $#%- 9'#9'-A#,8 :-*%-+ (3#*#5%+ 38#<%,'): " = u=2= n. 2 1. %5-&-2+ 49252 (3.32), &9#A#9, A' : ) 2#5-A#%-# ?T#9 (?- n 3-*-' 29#%<#%-H *5-%( *#-'#5%8 -%'#5, '.#. 252A<#%-H 'A%,'- "U ?) 2#5-A#%-# *#-'#5%. #+'%,'- 1 ; 3-*-' 2#5-A#%-H *5-%( *#-'#5%8 -%'#5, '.#. 29#%<#%-H 'A%,'- "U ) #,5- &*' 'A%,' (3#*#5%2H 38#<%,' -%'#5%. $#%-) " *#-'#5%2H #+'%,' 1 ; , ' -& ,'%<#%-+ " = p n u=2 (3:33) 9:% %.'- 9-%-95%(. ?T#9 (?-, '(. ?#,3#A-' &*%%2H 'A%,' (3:34) n = "12 u2=2 2: 69 2 2. 2,' X -9##' 3-&5%2H 42%$-H ,3#*#5#%-+ #+'%,'#. F (x). 2,' ':# *5+ X (35%+#',+ $#%'5%+ 3#*#5%+ '##9 '#-- #+'%,'#.. 8*, ,-52 6'. '##9(, 3- *,''A% ?5<9 ?T#9# (?- &% ,3#*#5#%-+ (?A%. ,''-,'-- u = (x ; MZx]) = x ?2*#' 3-?5-:#%% %95%(9. 6'9 ,52A# *#-'#5%(. -%'#5 *5+ MZX] ,52A.%. #5-A-%( X, 3#*#5#%%(. 3 4925# (3.32), ?2*#' 3-?5-:#%%(9. $#5H 3,'#%-+ 'A%8 *#-'#5%8 -%'#5 *5+ MZX] ,52A.%. #5-A-%( X, -9#HO#. 3-&5%(. &% ,3#*#5#%-+, %#?1*-9 &%' 5-? &% ,3#*#5#%-+ #+'%,'#. ,#*%#. -49#'-A#,. x = n;1 Pi xi, 5-? &% ,3#*#5#%-+ (?A%. ,''-,'-- u = (x ; MZx]) = x. %( ,3#*#5#%-+ 6'-1 #5-A-% &-,+' ' &% ,3#*#5#%-+ ,52A.%. #5-A-%( X. ' 5A-- - *#A- :*H',+ %# *-% A,'. D'' 4' ?(5 &9#A#% 2:# *%. 5+ ,&*%-+ 9'#9'-A#,-1 9*#5#., ,3,?%(1 3,5#*-' 3$#,,(, %?5H*HO##,+ ,,'# %*%,#5#%-+ ,'%(, %#?1*-9 ?5<. ,''-,'-A#,-. 9'#-5. 1874 8* 3 1900 8* _#.$-- *-5, 2644757 *#'#., '9 A-,5# 1359671 95A- - 1285086 *#A#. #+'%,' :*#%-+ 95A-? () ''-,'-A#,+ A,'' :*#%-+ 95A- & 2&%%(# 8*( ,,'-5 = 1359671=2644757 = 0 5141: *#, 9( -9##9 *#5 , 3,5#*'#5%,'H -,3('%-. , *29+ &9:%(9-,1*9- :*9, 3-A#9 A-,5 -,3('%-. n A#% #5-. 6'92 9:% -,35&' '##92 #%255-, ' - '##92 2-K35,. #+ -& %-1 2&(#' % ', A' #+'%,' 3+5#%-+ 95A- ?5-& 1/2 ( #+'%,' :*#%-+ *#A-). 9'#9'-A#,. 9*#5- A,'' :*#%-+ 95A- wn, %?5H*#9+ ,#*n *-<-1,+ *#'#., -9##' *-,3#,-H DZwn ] = p(1 ; p)=n. ,52 ?T#9 (?n A#% #5- (% #% 2644757) - #+'%,' p ?5-& 1/2, ' , (,. ,'#3#%H 'A%,'- DZwn] = (0 0003)2 . ,5- 3'-A#,- *,'#%(9- 2,5-',+ ,A-'' ,?('-+ , #+'%,'H %# 9#%## 0,95, ' -& $#%- (3.32) ,5#*2#', A' #+'%,' p :*#%-+ 95A- &5HA#% 9#:*2 ,5#*2HO-9- 8%-$9- : q q p > ; 1 96 DZwn] = 0 5135 p < + 1 96 DZwn] = 0 5147: %A#%-+ , ,'5 (,. 'A%,'H #+'%,'- p 3&5+#' 3-%+' & 3'-A#,- *,'#%#, A' 3 .%#. 9## '#A#%-# %#,5-1 5#', 3,5#*2HO-1 & 1900 8*9, A-,5 :*<-1,+ 95A- % :*(# 10000 *#'#. ,,'5+5 ' 5133 * 5145. 3&(H' 9%8A-,5#%%(# - 3*5:-'#5%(# %?5H*#%-+, #+'%,' p +5+#',+ *5% 2,'.A-. 1'#-,'-.. 70 3.8. 9 & # 2,' X ! 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N (0U 1). %5-&-2+ ,52A.%2H #5-A-%2 t, 3#*#5+#92H #%,'9 (3.38), 3-1*-9 (*2, A' 6' ,52A.%+ #5-A-% ,3#*#5#% 3 &%2 'H*#%' , = n ; 1 ,'#3#%+9- ,?*(. #+'%,' '8, A' ,52A.%+ #5-A-% t 33*#' -%'#5 Z;t=2n;1U t=2n;1 ], %1*-',+ 3 4925# (-,. 3.4) t=Z2 n;1 ! x ; a Pr ;t=2n;1 < s=pn < t=2n;1 = 2 f (t) dt: (3:39) 0 **-9 6'2 #+'%,' %. 1 ; . 8* -& #<#%-+ 2%#%-+ t=Z2 n;1 2 f (t) dt = 1 ; 0 (3:40) 9:% %.'- %'-5 ,3#*#5#%-+ 'H*#%' t=2n;1. 9#H',+ ,3#$-5%(# '?5-$( (,9. 3-5:#%-#), '(1 39#O#%( %'-5- t-,3#*#5#%-+, ,'#','2HO-# *#-'#5%. #+'%,'- 1 ; - ?T#92 (?- n. 71 0.4 0.3 f (t) 0.2 0.1 0.0 . ............................. ..... .... .... .... .... . .... . . .. ... . . .... .. . . ... . . . ... .. . ... . . . ... . . ... . .. ... . . . ... . . ... . . ... .. . ... . . ... .. . ... . . ... . . . ... .. . ... .. . ... . . ... . . . ... . . ... . . . .... . .... ...... . . . ... . . .... .... .. ... . .. .... . ..... ... ... ...... . . . ... ... ... ... .... . . .. ... .... ... ... ..... . . ...... ... ... ... ... ........ . . . .. .. .. . ... ... ... .... . . ... ... ... ... . ... ... ... .......... . . .. ... ... ... ... . ... ... ... ... ...... . . .... ... ... ... ... . ... ... ... ... ......... . . .. ... .. ... ... .. . . . . ... .... .... ... ... ............ . . .. ... ... ... ... ... ... . . . ... ... ... ... ... ... .......... . . ...... ... ... ... ... ... ... . . . . . . . . .. ... .. .. ... .. .... .................. ...... ... .. ... .. .. ... .. . . . . . . . . ... ... ... ... ... ... .. ... ... ...................... . . . . . . . . . . . . . . . ..................... .. .. .. .. .. .. .. .. .. .. ...................................... .... ... ... ... ... ... ... ... ... ... ... ... ... .. .. ... ... .... .... . . =2 ;3 ;2 =2 ;1 0 t ;t=2 n;1 1 2 3 t=2 n;1 3.4 | : 1 + + % , '1 2*#9 ,A-'' %'-5- t=2n;1 , ,'#','2HO-# &*%%. *#-'#5%. #+'%,'- 1 ; - ?T#92 (?- n, -&#,'%(9-. #?&2#9 %##%,', ,'+O## 5#. A,'- 2%#%-+ (3.39): p p Pr ;t=2n;1 s= n < (x ; a) < t=2n;1 s= n = 1 ; (3:41a) -5- p p Pr x ; t=2n;1 s= n < a < x + t=2n;1 s= n = 1 ; : (3:41b) & 6'8 p(:#%-+ *5+ #+'%,'('##', A' *#-'#5%(. -%'#5 p (x ; t=2n;1 s= n x + t=2n;1 s= n) %(#' %#-&#,'%# 9'#9'-A#,# :-*%-# , &*%%. #+'%,'H 1 ; . A%,' (3#*#5%+ 38#<%,') $#%9'#9'-A#,8 :-*%-+ ,5#*2HO+ : " = psn t=2n;1 : (3:42) 52A#%%(. *#-'#5%(. -%'#5 ?5*#' '#9- :# ,.,'9-, A' *#-'#5%(. -%'#5 *5+ 9'#9'-A#,8 :-*%-+ 3- -&#,'%9 . 2 . (<# '9#A5,, A' 3- %#8%-A#%%9 &,'%-- ?T#9 (?- ,3#*#5#%-# 'H*#%' ,'#9-',+ %95%92. 6'92 2:# 3- n 30 , $#5H 3,'#%-+ *#-'#5%(1 -%'#5 *5+ 9'#9'-A#,8 :-*%-+ 9:% 9#,' ,3#*#5#%-+ 'H*#%' -,35&' %95%# ,3#*#5#%-#. 6'9 ,52A# 3-?5-:#%%(# *#-'#5%(# -%'#5( *5+ 9'#9'-A#,8 :-*%-+ %1*+',+ 3 4925# (3.32), '. ,5#*2#' 3-%+' = s. 72 3.9. 9 & & 2,' ,52A.%+ #5-A-% X ! N (a ), 3-A#9 39#'( a - %#-&#,'%(. 5+ %1:*#%-+ 'A#A%(1 $#% 39#' a - -& 8#%#5%. %95%. ,23%,'- -&5#A#% (? ?T#9 n. 2,' % ,%%-- 6'. (?%.*#%( 'A#A%(# %#,9#O#%%(# $#%- %#-&#,'%(1 39#' v u n X u a^ = xU ^ = s = t n ;1 1 (xi ; x)2: (3:43) i=1 5+ 3,'#%-+ *#-'#5%8 -%'#5 *5+ ,#*%#8 *'-A#,8 '5%#%-+ ,,'-9 ,398'#5%2H ,52A.%2H #5-A-%2 2 2 = (n ; 21)s : (3:44) D' ,52A.%+ #5-A-% -9##' ,3#*#5#%-# 2 , n ; 1 ,'#3#%+9- ,?*(. #+'%,' '8, A' ,52A.%+ #5-A-% 2 33*#' -%'#5 (C1U C2), % (-,. 3.5) ZC2 2 2 2 Pr C1 < < C2 = f ( ) d : (3:45) C1 **-9 6'2 #+'%,' %. 1 ; . (?##9 &%A#%-+ C1 - C2 -& 2,5-. 2 Z1 2 2 Pr > C2 = f ( ) d = 2 (3:46a) C2 ZC1 f (2) d2 = 2 : (3:46b) 0 -,. 3.5 ,'#','2HO-# *# -5-%#.%(# '3#$--, :*+ , 35O*H =2, &<'-1%(. #<+ (%3-9#, A-,5#%%(9- 9#'*9-) 2%#%-+ (3.46), %1*+' %'-52 -,3#*#5#%-+ C1 = 21;=2 n;1 - C2 = 2=2 n;1 . 2*#9 *5%#.<#9 ,A-'' %'-5- 2-,3#*#5#%-+ -&#,'%(9-. #?&2+ 49252 ! 2 ( n ; 1) s 2 2 Pr 1;=2 n;1 < < =2 n;1 = 1 ; 2 -9##9 1 0 v v u u u n ; 1 n ; 1 u A = 1 ; < < s t 2 (3:47) Pr @s t 2 Pr 2 < C1 = =2 n;1 1;=2 n;1 73 0.10 f (2) 0.05 ...................... ........ ......... ..... ...... ..... .... . . . ..... .. . . ..... ... .... . . .... . . . .... . . .... . . . .... . . . .... .. .... . . . .... . .... .. . .... . . .... .. . .... . . . .... . . .... . . .... . . . ..... . . ..... . . . ..... . . . ...... . . ..... . . . ..... . . ..... . . ...... .. . ...... . ....... . ...... ............ . . .. . ... ................ . . ... . ... ... ................ . .... ... . ... ... ... ... .............. . . ... ... . .. .. ... .. ... ................ . . .... .... .... ... ... ... ... ... ... ... ......................... . . . .. .. .. . ... ... ... ... ... ... ... ... ... .............................................. . . . . . . . . .... ... .. ... .. ... .. ... .. .. .. ... ... .. .................................................................... .......................... ... ... ... ... ... .. .. .. .. ... ... . . =2 =2 0.00 0 5 10 2 C1 = 21;=2 n;1 15 20 25 C2 = 2=2 n;1 3.5 | : + + -5-, ?5## ' : Pr (s 1 < < s 2 ) = 1 ; (3:48) 8*# v v u u u n ; 1 t t 2n ; 1 : 2 = u (3:49) 1 = 2 1;=2 n;1 =2 n;1 644-$-#%'( 1, - 2, ,'#','2HO-# *#-'#5%. #+'%,'- 1 ; - A-,52 ,'#3#%#. ,?*( n, 39#O#%( 3-5:#%--. 2 . 3- n ! 1 ,3#*#5#%-# 2 3-?5-:#',+ %95%92, ' 3- *,''A% ?5<9 ?T#9# (?- (n 50) *#-'#5%(. -%'#5 9:% %.'- 3 4925# ! s s p < < p Pr = 1 ; (3:50) 1 + u=2= 2n 1 ; u=2= 2n 8*# u=2 | %'-5 ,'%*'-&%%8 %95%8 ,3#*#5#%-+, ,'#','2HO-. *#-'#5%. #+'%,'- 1 ; . 3.10. ' 3.1 2,' ,52A.%+q#5-A-% X ,3#*#5#% 3 %95%92 &%2 , 39#'9- MZX] = a - = DZX], -5-, ?5## ', 32,' X ! N (a ). #?2#',+ 3 #&25''9 %?5H*#%-. x1 x2 : : : xn $#%-' 39#'( a - - %.'- $#%- ,-99#'-- - 6,$#,,. 74 () -9#%++ 9#'* 99#%', -9##9 8 Pn > < 1 = 1 ) MZX^ ] = a^ = x^ = n1 i=1 xi n > : 2 = 2 ) DZX^ ] = x2 = s2 = n1 P (xi ; x^)2 i=1 () 8*# X1, X2, : : :, Xn 3#*,'5+H' ?( n 6&#935+ ,52A.%. #5-A-%( X , *%-9 - '#9 :# 9'#9'-A#,-9 :-*%-#9 a. :#9, A' ,#*%## -49#'-A#,# %?5H*#9(1 &%A#%-. +5+#',+ ,,'+'#5%., %#,9#O#%%. - 644#'-%. $#%. 9'#9'-A#,8 :-*%-+. ,'+'#5%,' $#%- ,5#*2#' -& '##9( #%255 m ; p < " = 1 (" > 0): lim Pr n!1 n & '##9( #%255- ,5#*2#', A' ,#*%## -49#'-A#,# +5+#',+ ,,'+'#5%. $#%. 9'#9'-A#,8 :-*%-+ 3- 5H?9 &%# ,3#*#5#%-+. .*#9 9'#9'-A#,# :-*%-# ,#*%#. -49#'-A#,. " X # "X # n n n X 1 1 1 MZx] = M n Xi = n M Xi = n MZXi] = n1 na = a: i=1 i=1 i=1 5#*'#5%, ,#*%## -49#'-A#,# +5+#',+ %#,9#O#%%. $#%. 9'#9'-A#,8 :-*%-+. ,,5#*2#9 644#'-%,' $#%- a^ = x^, ,A-'+ -&#,'%(9. 5+ 6'8 (A-,5-9 *-,3#,-H ,#*%#. -49#'-A#,. " X # "X # n n n 2 X 1 1 1 1 2 DZx] = D n Xi = n2 D Xi = n2 DZXi] = n2 n = n : i=1 i=1 i=1 .*#9 %-:%HH 8%-$2 %##%,' -9#. 5+ 6'8 (A-,5-9 " 2 ( " 2 2 !)# 2# @ 1 @ ( x ; a ) ( x ; a ) M @a2 ln p exp ; 2 2 = ;M @a2 2 2 = ; 12 : 2 %-:%++ 8%-$ %##%,' -9# 3- N = n ,3*#' , *-,3#,-#. ,#*%#. -49#'-A#,. 2 ; n M Z @ 2f (1x )=@p2] = n ' ,#*%## -49#'-A#,# 3- -&#,'%9 +5+#',+ 644#'-%. $#%. 9'#9'-A#,8 :-*%-+. :#9, A' $#% *-,3#,--, (A-,5+#9+ 3 4925# (), +5+#',+ ,9#O#%%.. .*#9 9'#9'-A#,# :-*%-# 693--A#,. *-,3#,--, 3#*-'#5% 3#?& 49252 (): n n X X 1 1 2 2 s = n (xi ; x) = n Z(xi ; a) ; (x ; a)]2 = i=1 i=1 75 # "X n n X 1 2 2 = n (xi ; a) ; 2 (x ; a) (xi ; a) + n (x ; a) = i=1 i=1 n X = n1 (xi ; a)2 ; (x ; a)2 U i=1 i i h X h MZs2] = 1 M (Xi ; a)2 ; M (x ; a)2 = n i 2 !23 "X # X 1 1 = 2 ; M 4 n (Xi ; x) 5 = 2 ; n2 M (Xi ; a)2 = i i 2 1 2: = 2 ; nn 2 = n ; n 2 2 MZs ] 6= , ' $#% *-,3#,--, 3#*#5+#9+ 3 4925# (), +5+#',+ ,9#O#%%.. 9#O#%-# $#%- 2,'%+#',+ 29%:#%-#9 #E % 9%:-'#5 n=(n ; 1). - *,''A% ?5<-1 n (n > 30) ,9#O#%-# $#%- %#&%A-'#5%. - 959 ?T#9# (?- (n 30) %#,9#O#%%+ $#% (A-,5+#',+ 3 4925# n X DZX] = ^X2 = s2 = n ;1 1 (xi ; x)2 : i=1 -#*#9 ?#& *&'#5,' %#'(# #&25''( '%,-'#5% ,.,' $#% *-,3#,--. ,5- 39#'( %95%8 &% %#-&#,'%(, ' $#% *-,3#,--, 3#*#5+#9+ 3 4925# (), %# ?5*#' ,.,'9 644#'-%,'-. *% $#% P 1 n 2 ^ = n i=1 (xi ; a)2, #,5- -&#,'% 9'#9'-A#,# :-*%-# a, ?5*#' ,.,'9 %#,9#O#%%,'-, ,,'+'#5%,'- - 644#'-%,'-. $#%- ,-99#'-- A^ - 6,$#,, E^ , +5+HO-#,+ 42%$-+9- $#%'5%(1 99#%', %1*+',+ 3 49259 : n 3 = 3 ) A^ = 3 = 1 X 3 ( x i ; x) U 3 3 3 3 ns i=1 n 4 ; 3 = 4 ; 3 ) E^ = 4 ; 3 = 1 X 4 4 4 4 ns4 i=1 (xi ; x) ; 3: ' 3.2 &', A' '%,-'#5%+ A,'' m=n 3+5#%-+ ,?('-+ A 3- n -,3('%-+1 #%255- +5+#',+ ,,'+'#5%., %#,9#O#%%. - 644#'-%. $#%. #+'%,'- p. () ,'+'#5%,' $#%- ,5#*2#' -& '##9( #%255- : m ; p < " = 1 (" > 0): lim Pr n!1 n 76 3#*#5-9 9'#9'-A#,# :-*%-# '%,-'#5%. A,''(. 2*#9 358', A' m = X, '. #. A,''2 3+5#%-+ ,?('-+ A n -,3('%-+1 ,1#9( #%255,,9'-#9 ,52A.%2H #5-A-%2 X, ,3#*#5#%%2H 3 ?-%9-5%92 &%2 " # M Xn = n1 MZX] = n1 np = p: M Zm=n] = p, ' '%,-'#5%+ A,'' m=n +5+#',+ %#,9#O#%%. $#%. #+'%,'- p. 3#*#5-9 *-,3#,-H '%,-'#5%. A,''( " # m D n = D Xn = n12 DZX] = n12 npq = pq n: .*#9 %-:%HH 8%-$2 %##%,' -9#, 2A-'(+, A' f (x ) = f (X = xU p) = CnX pX (1 ; p)n;X : 8* " " 2 # 2 ln f (xU p) # @ @ X M = M @p2 ln Cn + X ln p + (n ; X) ln(1 ; p) = @p2 # " n ; X n ; np = ; n : X ; = M ; p2 ; (1 ; p)2 =; np 2 p (1 ; p)2 pq 5#*'#5%, 358+, A' *5+ $#%- #+'%,'- ,*#5% *-% 6,3#-9#%' (n = 1), %.*#9, A' %-:%++ 8%-$ %##%,' -9# % ; n M Z @ 2f (1x )=@p2] = pqn : %-:%++ 8%-$ %##%,' -9# ,3*#' , *-,3#,-#. '%,-'#5%. A,''(, ' '%,-'#5%+ A,'' +5+#',+ 644#'-%. $#%. #+'%,'- p. ' 3.3 52A.%+ #5-A-% X ,3#*#5#% 3 3&'#5%92 &%2 , 35'%,'H #+'%,'#. f (xU ) = exp(;x) , 39#'9 . #?2#',+ 3 #&25''9 %?5H*#%%(1 &%A#%-. x1 x2 : : : xn 6'. ,52A.%. #5-A-%( %.'- $#%2 ^ 39#' . () 3-<#9 42%$-H 3*3*?-+ L = exp(;x1) exp(;x2) : : : exp(; xn) = 77 n exp n ! X ; xi : i=1 K8-49-2+, -9##9 ln L = n ln ; n X i=1 xi: -44##%$-2+ 3 39#'2 , %1*-9 n @ ln L = n ; X @ i=1 xi: -%-+ 3-&*%2H %25H, 352A#9 n n ;X n = 1=x: x i = 0 ) = Pn i=1 i=1 xi ' 3.4 2,' ,52A.%+ #5-A-% X ,3#*#5#% 3 &%2 2,,% k Pr(X = k) = k! exp(;) k = 1 2 : : : : #?2#',+ % ,%%-- %?5H*#%%(1 &%A#%-. x1 = k1, x2 = k2, : : :, xn = kn , $#%-' %#-&#,'%(. 39#' 6'8 ,3#*#5#%-+. () 3-<#9 42%$-H 3*3*?-+ x2 xn x1 L = x ! exp(;) x ! exp(;) : : : x ! exp(;) = 1 2 n n S X = x !x ! : : : x ! exp(;n) S = xi: 1 2 n i=1 5+ %1:*#%-+ $#%- , '.#. '8 &%A#%-+ = (x1 x2 : : : xn), 3'9 42%$-+ 3*3*?-+ ?O#',+ 9,-929, 2*?% 3##.'- ! # n n X X ln L = ;n + xi ln ; ln(xi!) ,5#*'#5%, i=1 i=1 n @ ln L = ;n + 1 X @ i=1 xi: -%-+ 3-&*%2H %25H, -9##9 n X ;n + 1 xi = 0 i=1 A' *#' n ^ = 1 X xi = x: n i=1 78 ' 3.5 52A.%+ #5-A-% X ,3#*#5#% 3 %95%92 &%2. #?2#',+ 3 #&25''9 %?5H*#%%(1 &%A#%-. x1 x2 : : : xn 6'. ,52A.%. #5-A-%( $#%-' 39#'( a - %95%8 &%. () 5'%,' ,3#*#5#%-+ #+'%,'#. %95% ,3#*#5#%%. ,52A.%. #5-A-%( % 2! 1 ( x ; a ) f (xU a ) = p exp ; 2 2 2 ,5#*'#5%, 42%$-+ 3*3*?-+ -9##' -* ! n X 1 1 2 exp ; 2 2 (xi ; a) : L = p ( 2 )n i=1 3-<#9 58-49-A#,2H 42%$-H 3*3*?-+ n X ln L = ; n2 ln(2) ; n ln ; 2 1 2 (xi ; a)2: i=1 -44##%$-2+ 58-49-A#,2H 42%$-H 3*3*?-+ 3 a - , -9##9 n @ ln L = 1 X @a 2 i=1(xi ; a) = 0 n @ ln L = ; n + 1 X 2 @ 3 i=1(xi ; a) = 0 '2* %1*-9 n n X X ^ 2 = s2 = n1 (xi ; x)2: a^ = n1 xi = xU i=1 i=1 D'- $#%- ,35- , $#%9-, 352A#%%(9- 9#'*9 99#%'. ' 3.6 2,' X -9##' %95%# ,3#*#5#%-# , -&#,'%(9 ,#*%-9 *'-A#,-9 '5%#%-#9 = 2. #?2#',+ %.'- *#-'#5%(. -%'#5 *5+ 9'#9'-A#,8 :-*%-+ a, #,5- ,#*%## -49#'-A#,# &%A#%-# #&25'' n = 16 3+9(1 %'A%(1 -&9##%-. x = 20 09. #-'#5%2H #+'%,' (1 ; ) 3-%+' %. 0,90. () 5&2+, '?5-$9- 42%$-- K35,, 3 &*%%. #+'%,'- (1 ; ) %1*-9 u=2 = u005 = 1 64: .*#9 'A%,' (3#*#5%2H 38#<%,') $#%" = 1 64 p2 = 0 82: 16 5#*'#5%, -,9(. *#-'#5%(. -%'#5 ,,'5+#' Z20 09 ; 0 82U 20 09 + 0 82], -5- Z19 27U 20 91]. 79 9(,5 352A#%%8 #&25'': #,5- ?2*#' 3-&#*#% *,''A% ?5<# A-,5 (? *%%8 ?T#9, ' 90 % -& %-1 *#-'#5%(# -%'#5( %H' 9'#9'-A#,# :-*%-# - '5 10 % ,52A# $#%-#9# 9'#9'-A#,# :-*%-# 9:#' (.'- & 8%-$( *#-'#5%(1 -%'#5. ' 3.7 29#%+ ,52A.%+ #5-A-% (X Y) ,3#*#5#% 3 %95%92 &%2 , 35'%,'H #+'%,'#. f (x y) = 2 1p1 ; 2 X Y ( 2 2 (x ; m )(y ; m ) (y ; m )2 ! ) 1 ( x ; m X) X Y exp ; 2(1 ; 2) + 2 Y : 2 ; X X Y Y #?2#',+ 3 #&25''9 %?5H*#%-. *29#%. ,52A.%. #5-A-%( (Xi Yi), 8*# i = 1 2 : : : n, $#%-' 39#'( mX , mY , X , Y - . () -9#%++ 9#'* 99#%', -9##9 n X 10 = 10 ) m^ X = x = n1 xiU i=1 n X 01 = 01 ) m^ Y = y = n1 yiU i=1 n X 20 = 20 ) ^X2 = s2X = n1 (xi ; x)2 U i=1 n X 02 = 02 ) ^Y2 = s2Y = n1 (yi ; y)2 U i=1 n 11 ) ^ = 1 1 X X Y n X Y i=1 (xi ; x) (yi ; y) : ' 3.8 2,' x1 = 2 015U x2 = 2 020U x3 = 2 025U x4 = 2 020U x5 = 2 015 | #&25''( %#&-,-9(1 %'A%(1 -&9##%-. '5O-%( 9#'55-A#,. 35,'-%-. #?2#',+ : 1. $#%-' , 39OH *#-'#5%8 -%'#5 -,'-%%2H '5O-%2 (9'#9'-A#,# :-*%-#) 35,'-%-. #-'#5%2H #+'%,' 1 ; 3-%+' %. 0,95. 2. .'- 9-%-95%# A-,5 -&9##%-., '(# %* (35%-', A'?( , %*#:%,'H 1 ; = 0 95 9:% ?(5 2'#:*', A' 3#*#5%+ 38#<%,' 'A#A%. $#%- -,'-%%. '5O-%( 9#'55-A#,. 35,'-%- %# 3#(<#' 0,95. () 2*#9 ,A-'' #&25''( -&9##%-+ %?5H*#%%(9- &%A#%-+9- ,52A.%. #5-A-%( X, '+ ,3#*#5#% 3 %95%92 &%2 , %#-&#,'%(9- 39#'9- a - . .*#9 'A#A%(# $#%- 6'-1 39#'. 80 9# %?5H*#%-+ 1 2 3 4 5 xi 2,015 2,020 2,025 2,020 2,015 xi ; x {0,004 +0,001 +0,006 +0,001 {0,004 (xi ; x)2 0,000016 0,000001 0,000036 0,000001 0,000016 5#*'#5%, â = x = 105095 = 2 019U s ^ = s = 0 00007 = 0 0042: 4 '?5-$9 ,3#*#5#%-+ 'H*#%' (,9. 3-5:#%-#) 3 *#-'#5%. #+'%,'- 1 ; = 0 95 - A-,52 ,'#3#%#. ,?*( = n ; 1 = 4 %1*-9 %'-5 ,3#*#5#%-+ t00254 = 2 776. 5#*'#5%, 3#*#5%+ 38#<%,' 'A#A%. $#%" = psn t=2n;1 = 0p0042 2 776 = 0 0052: 5 #-'#5%(. -%'#5 #% Zx ; "U x + "] = Z2 0138U 2 0242]: 5+ ,A#' 9-%-95%8 A-,5 -&9##%-., %#?1*-9(1 *5+ 3#*#5#%-+ -,'-%%. '5O-%( 35,'-%, 38#<%,'H, %# 3#(<HO#. 0,003, 3-9#%-9 p 49252 " = st=2n;1 = n. 5#*'#5%, 2 0 000018 n ";2t2=2n;1s2 = 2 7760 000009 16 : -9 ?&9, *5+ '8 A'?( 3#*#5%+ 38#<%,' " -,'-%%8 &%A#%-+ '5O-%( 35,'-%- %# 3#(<5 " = 0 003, ,5#*2#', 39-9 5 3*#5%%(1 -&9##%-., (35%-' #O# (n ; 5) = 16 ; 5 = 11 -&9##%-.. ' 3.9 ( 2) 9##',+ (? ?T#99 N . 5+ %#E %.*#%( (?A%# ,#*%## xN (?A%+ *-,3#,-+ DN . 52A#% #O# *% &%A#%-# xN +1. ,,9'-#',+ ?T#*-%#%%+ (? ?T#99 N + 1. #?2#',+ (&-' (?A%(# $#%- xN +1 - DN +1 A##& 2:# -9#HO-#,+ $#%- xN , DN - %# &%A#%-# xN +1. () 5+ (?A%8 ,#*%#8 -9##9 ! X NX +1 N 1 1 xi + xN +1 = N 1+ 1 (NxN + xN +1) : (1) xN +1 = N + 1 xi = N + 1 i=1 i=1 .*#%%# (?A%# ,#*%## xN +1 -,35&2#9 *5+ %1:*#%-+ (?A%. *-,3#,-NX +1 2 1 DN +1 = N + 1 xi ; xN +1 : (2) i=1 81 285(1 ,?1 3* &%9 ,299-%-+ (A'#9 - *?-9 (?A%# ,#*%## xN . 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U 5'#%'-%2H 8-3'#&2 a 6= a Ha : 1 6= 2 1 2 | % ! % . ''-,'-A#,-# 39#'-A#,-# 8-3'#&( 982' ,*#:' *% -5- %#,5 3#*35:#%-. '%,-'#5% 39#' ,3#*#5#%-+ 42%$--. 5. 9#'-A#,+ 8-3'#& %&(#',+ , #,5- ,*#:-' '5 *% 3#*35:#%-# '%,-'#5% 39#'. 3-9#, #,5- a | 9'#9'-A#,# :-*%-# %95% ,3#*#5#%%. ,52A.%. #5-A-%(, ' 8-3'#& fH0 : a = 0g | 3,'+. 6. 9#'-A#,+ 8-3'#& %&(#',+ 1 , #,5- % ,,'-' -& %#A%8 -5- ?#,%#A%8 A-,5 3,'(1 8-3'#&. 3-9#, #,5- a | 9'#9'-A#,# :-*%-# %95% ,3#*#5#%%. ,52A.%. #5-A-%(, ' 8-3'#& Ha : a > 0 5 +5+#',+ ,5:%., ' % ,,'-' -& ?#,%#A%8 9%:#,' 3,'(1 8-3'#& -* fHa : a = aig, 8*# ai | 5H?# &*%%# A-,5, ?5<## 0,5. 90 4.2. & # ,''-,'-A#,-1 8-3'#& ,2O#,'5+#',+ % ,%# *%%(1 (?-. 5+ 6'8 -,35&2H' ,3#$-5%(9 ?&9 3*?%%2H ,52A.%2H #5-A-%2 ((?A%2H ,''-,'-2), +5+HO2H,+ 42%$-#. %?5H*#%%(1 &%A#%-., 'A%# -53-?5-:#%%# ,3#*#5#%-# '. -&#,'%. D'2 (?A%2H ,''-,'-2 ?&%AH' &5-A%(9- ?29- &-,-9,'- ' &% #E ,3#*#5#%-+, %3-9#, u, #,5- % ,3#*#5#% 3 %95%92 &%2, F | #,5- % -9##' ,3#*#5#%-# J-<#, t | #,5- % -9##' ,3#*#5#%-# 'H*#%'. *%%9 384# $#5+1 ?O%,'- ?2*#9 ?&%A' #E A##& K . 1. ' () K %&(H' ,52A.%2H #5-A-%2 K, , 39OH '. 3-%-9H',+ #<#%-+ 3-%+'-- -5- '5%#%-(*-%2'. %25#. 8-3'#&(. 5+ 3#- %25#(1 8-3'#& 3 (?A%(9 *%%(9 (A-,5+H' A,'%(# &%A#%-+ 1*+O-1 -'#-. #5-A-% -, '-9 ?&9, 352AH' A,'%# (%?5H*#9#) &%A#%-# -'#-+. 2,' *5+ 3#- %#'. %25#. 8-3'#&( H0 '%,-'#5% 39#' ,3#*#5#%-+ ,52:-' (?A%+ ,''-,'- (-'#-.) K. #*35:-9, A' 35'%,' ,3#*#5#%-+ #+'%,'#. (?A%. ,''-,'-- K 3- 2,5-,3#*5-,'- 3#+#9. 8-3'#&( H0 % f (K jH0), 9'#9'-A#,# :-*%-# ,''-,'-- K % K0 . 8* #+'%,' '8, A' ,52A.%+ #5-A-% K 33*#' 3-&5%(. -%'#5 ZK1;=2U K=2], 9:% %.'- 3 4925# (-,. 4.1) Pr K1;=2 < K < K=2 = KZ=2 K1;=2 f (K jH0 ) dK: (4:1) **-9 6'2 #+'%,' %. 1 ; - (A-,5-9 %'-5- K1;=2 - K=2 35'%,'- ,3#*#5#%-+ f (K jH0) 3- 2,5-+1 : Pr K K1;=2 = Pr K K=2 = K1Z;=2 ;1 Z1 K=2 f (K jH0) dK = 2 f (K jH0) dK = 2 : (4:2a) (4:2b) 5#*'#5%, #+'%,' '8, A' ,52A.%+ #5-A-% K ?2*#' %1*-',+ %2'- -%'#5 ZK1;=2U K=2], % 1 ; . '#','#%% #+'%,' '8, A' ,52A.%+ #5-A-% K ?2*#' %1*-',+ %# 6'8 -%'#5, % . **-9 #+'%,' %,'5 95., A'?( 33*%-# ,52A.%. #5-A-%( K & 3#*#5( -%'#5 ZK1;=2U K=2] 9:% ?(5 ,A-'' 95#+'%(9 ,?('-#9. 8*, -,1*+ -& 3-%$-3 3'-A#,. %#&9:%,'- 95#+'%(1 ,?('-., 9:% ,A-'', A' #,5- %25#+ 8-3'#& ,3#*5-, ' 3- #E 3## 91 0.6 f (K jH0) 0.4 0.2 0.0 ?5,' *32,'-9(1 &%A#%-. K ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . ... ... .......................... . . . . . . . ... . ... ....... .. . . . . ... ... . . ..... .... . ... ... . . . ..... . . . ... ... . . .... .. ... . ... . . . . ..... . ... ... . . . .... ... .. ... . . . .... ... . ... . . . .... ... . ... . . . . . ... ... .... .. . . ... . .... ... ....... .... ..... ... ... ... ... .... .. ... .... ....... ........ ..... . ........ ..... ......... ... ..... . . . . ... ....... ..... .... . . . . ... .... ...... .. ... ... . . . ... ... ......... . .. ... ... ... . . ... ... ... ........ . . ...... .... .... .... . . ... ... ... ... ....... . . . .... ... .. .. .. . . . ... ... ... ... ............. . . . . ...... ... ... ... ... ... . . ... ... ... ... ... ................. . . . . . . . ....... ... ... ... ... ... ... . . ... ... ... ... ... ... .. ...................... . . . . . . . . . . . . . . . ... .. .. .. ... ... ... .. ... ......................................... ............................................ ... ... .. ... .. .. .. .. ... ... ... .. . ... ... .. . .. ... .. . -'-A#,+ ?5,' -'-A#,+ ?5,' 5O* = =2 ;3 5O* = =2 ;2 ;1 K1;=2 0 K 1 2 3 K=2 4.1 | J / / + K ( ) 0.4 f (K jH0) 0.2 0.0 ............ ................ ........................ ........ ........ ........ ....... . . . . . ...... ..... ...... ...... . . . ..... . . . . . ..... . .. . . ..... . . ... ..... . . . . ...... .. . . . . . ........... .. . . . . . . .... .... ............... . . . . . . ... . . ... .... ............... . . . . . . ..... . . . . . .... ... .... .... ..................... . . . . . .... ... ... ... ... ... ... ............................. .................... .................................................... .. . . . . .. . . ............................................ ... .... .... .. ... .. ;3 ;2 ;1 0 K 1 5O* = 2 3 K 4.2 | : #1 K 0.4 f (K jH0) 0.2 0.0 ....................................... ........... ........ ........ ........ ....... ...... . . . . . . ...... .... . . ...... . . .. . . ..... . . .. . ..... . . . .. ..... . . . . ...... ... . . . . . ...... . . . . . . ....... . . ... .... . . . ........ . . . . ... ... ... . ......... . . . . . . . .. .. ... ... .......... . . . . . . . . . . . . ........... ...... ... ... ... ... . . . . . . . . .................. . . . . . . ........ ... ... .. ... ... ... . . . . . . . ................................. . . . . . . . . . . . . . . . . . . . . . . ........................ ...................................... . . . . . . . . . ... ... .... . ... .. 5O* = ;3 ;2 ;1 K1; 0 K 1 2 3 4.3 | : #1 K 92 , 39OH -'#-+ K 3 *%%(9 *%. (?- %?5H*#9# (A,'%#) &%A#%-# -'#-+ K *5:% ?+&'#5% 33,' -%'#5 ZK1;=2U K=2]. ,5- :# %?5H*#9# (A,'%#) &%A#%-# -'#-+ K 33*#' & 3#*#5( -%'#5 ZK1;=2U K=2], ' 3-&.*#' 95#+'%#, 3'-A#,- %#&9:%# ,?('-#, '.#. ,A-'#',+, A' , #+'%,'H (1 ; ) 3#+#9+ %25#+ 8-3'#& %# ,3#*5-. 6'92 ?5,' ZK1;=2U K=2] %&(H' $ % ,52A.%. #5-A-%( K, 3- '(1 %25#+ 8-3'#& %# '5%+#',+, ?5,'Z;1U K1;=2] - ZK=2U 1] | ! -5- -'-A#,. ?5,'H -'#-+ K. ,5- -'-A#,-# ?5,'- ,358H',+ ,3 - ,5# ' 9'#9'-A#,8 :-*%-+ ,52A.%. #5-A-%( K, ', 6' -&?:#% % -,. 4.1, ' -'-A#,+ ?5,' %&(#',+ , -'#-. K . %#'(1 ,52A+1 6,3#-9#%'' ?(#' '#* 2?#:*#%, A' K > K0 -5K < K0. 6'9 ,52A# -'-A#,-# ?5,'- +5+H',+ *%,'%%-9-. &9:%( - -'-A#,-# ?5,'-. -,. 4.2 - 4.3 -&?:#%( 3,'%%++ - 5#,'%%++ -'-A#,-# ?5,'- ,'#','#%%. 2 . -'-A#,+ 'A K ( K - ), '*#5+HO+ ?5,'- 3-%+'-+ -5- '5%#%-+ 3#+#9. %25#. 8-3'#&(, *5+ -'#-+ , 3,'%%#. -'-A#,. ?5,'H %1*-',+ -& 2,5-+ Z1 Pr (K K) = f (K jH0 ) dK = : (4:3) K D' :# -'-A#,+ 'A, '*#5+HO+ ?5,'- 3-%+'-+ -5- '5%#%-+ 3#+#9. %25#. 8-3'#&(, *5+ -'#-+ , 5#,'%%#. -'-A#,. ?5,'H %1*-',+ -& 2,5-+ KZ1; Pr (K K1;) = f (K jH0) dK = : (4:4) ;1 ', 9:% ,4925-' 3$#*22 3#- 39#'-A#,-1 8-3'#& , 39OH -'#-# &%A-9,'- ,5#*2HO-9 ?&9. 2. # 8-3'#&( , 39OH ,''-,'-A#,8 -'#-+ &%A-9,'- #,' 3-5 '5%#%-+ %25#. 8-3'#&(. D' 3-5 &5HA#',+ &?-#%-- ?5,'&9:%(1 &%A#%-. ,52A.%. #5-A-%( K % *# %#3##,#HO-#,+ 3*?5,'-. - 6'9 %25#+ 8-3'#& '#8#',+, #,5- %?5H*#9# (A,'%#) &%A#%-# -'#-+ K 3-%*5#:-' -'-A#,. 3*?5,'-, - ,A-'#',+ ,85,2HO#.,+ , 3('9, #,5- %?5H*#9# &%A#%-# -'#-+ K %# 3-%*5#:-' -'-A#,. 3*?5,'-. 93 4.3. :*, & . < & - 3## ,''-,'-A#,-1 8-3'#& 3 (?A%(9 *%%(9 ,#8* ,2O#,'2#' -, 3-%+'-+ 5:%8 #<#%-+. D' ?T+,%+#',+ '#9, A' ?T#9 (?- %#A#%, - 36'92 %#5&+ 'A% 3#*#5-' %- -* 42%$-- F (x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f (K jHa) | 35'%,' ,3#*#5#%-+ (?A%. ,''-,'-- K 3- 2,5--, A' 5'#%'-%+ 8-3'#& Ha +5+#',+ #%., ' #+'%,' ,#<#%-+ <-?'8 * *5+ -'#-+ , 5#,'%%#. -'-A#,. ?5,'H 9:% (A-,5-' 3 4925# Z1 = Pr (K K1;) = f (K jHa ) dK: (4:5) K1; 4. ; $ M K %&(#',+ #+'%,' (1 ; ) %#,#<#%-+ <-?- '8 * (9O%,' -'#-+ K | 6' #+'%,' '5%#%-+ %##%. 8-3'#&( Ha), '. #. M = 1 ; . -,. 4.4 *% 8#9#'-A#,+ -%'#3#'$-+ #+'%,'#. <-? 3#8 *, '8 * - 9O%,'- -'#-+ K, -9#HO#8 5#,'%%HH -'-A#,2H ?5,'. & -,. 4.4 -*%, A', 3##*-8+ %'-5 K1; 5# (29#%<+ <-?2 3#8 *), 9( '#9 ,9(9 2#5-A-#9 <-?2 '8 *. :% 3&', A' #+'%,' ,#<#%-+ <-?- '8 * +5+#',+ 42%$-#. %#,5-1 3##9#%%(1 : A-,5 -&9##%-. n, 2%+ &%A-9,'- , 1'# 5'#%'-%. 94 0.2 f (K jHi) 0.1 0.0 f (K jHa) f (K jH0) ........................... .............................. ...... ..... ..... .... ..... .... ..... .... .... .... ... ..... . . . . . .... .. .. ...... . . . . . ... .... . . . . . . . . .... . . . ............. . . . . . . .... . . . ... .... .. . . . . ... . ... .. ... . . ... . . . .......................... ... ... . . . ... ... . .... . . . . . ... .. .................................. . . ... . .... . . . . ... . . . ... ... ... .. ... . . .......................................... ... .. . . . .... .... .. ........ . . .... .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... . .... ..... . . . . . .... . . .. .............................................................. .. . . .... . . . . ..... ..... .... ..... .. . . . . . . . .... . ... ... ..... . . . . . . . . . . . . ..... .. .... ... ............................................................................. . . . . . . . . ....... . . ....... .. .... ... ... ... . . . . . . . . ......... . . . . . . . . .......... ... ... .. ... ... ... . . . . . . ............ . . . . . . . . . . . . . . . . ....... ... ... ... ... ....................................................................................................................... . ................. . . . . ............ . . . . . . . . . . . . . . . . . . . . . . .................................................................. ........................................................................................... . . .. . .. . .. ... ... .... . ... ... ... .. ;4 5O* = ;2 0 K1; 2 K 5O* = 4 6 8 4.4 | Q0 + + K , 12+ 1 1 8-3'#&( Ha, 3-9#%+#98 -'#-+ K, '. #. = f (n Ha K ): - 6'9 (35%+H',+ ,5#*2HO-# 3#*#5%(# ,'%<#%-+ : nlim !1 nlim !1 = 0U (4:6) = 0U (4:7) lim = 1: (4:8) !0 #%,' (4.6) - (4.7) 8+' '9, A' ,''-,'-A#,-# *&'#5,' -,'-%%,'- 8-3'#& H0 - Ha ,'%+',+ *,'#%(9- '5 3- ?#,%#A% ?5<9 ?T#9# (?-. 9# '8, -& 6'-1 3#*#5%(1 ,'%<#%-. ,5#*2#', A' #*-%,'#%%(9 ,3,?9 *%#9#%%8 29#%<#%-+ #+'%,'#. <-? 3#8 - '8 * +5+#',+ 2#5-A#%-# ?T#9 (?-. #%,' (4.8) 8-' '9, A', 29#%<+ #+'%,' <-?- 3#8 * * %25+, 9( 3- 4-,-%%9 ?T#9# (?- *32,#9 %#8%-A#%%(. -, ,*#5' <-?2 '8 *. :# ,5#*2#' (?-' 2#% &%A-9,'- ,''-,'-A#,-1 -'#-#? 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() #*35:-9, A' #&25''( -&9##%-. 3A%,'- % ,:'-# '#1 9 ?#'% 3*A-%+H',+ %95%(9 -5- 3-?5-:#%% %95%(9 ,3#*#5#%-+9. 85,% 2,5-H, %9 %#?1*-9 3#-' %25#2H 8-3'#&2 fH0 : 1 = 2 = 3g 3'- 5'#%'-%. 8-3'#&( fHa : & 1 g. ?T#*-%-9 ,398'#5%(# (A-,5#%-+, %#?1*-9(# *5+ ,A#' (?A%. ,''-,'-- 2 3 4925# (4.34), '?5-$2. } x1i x2i x3i (x1i ; x1)2 (x2i ; x2)2 (x3i ; x3)2 1 195 215 201 36 121 36 2 200 201 204 1 9 9 3 204 202 221 9 4 196 4 205 198 210 16 36 9 5 201 | 199 0 | 64 299( 1005 816 1035 62 170 314 ',H*: x1 = 201U x2 = 204U x3 = 207U s21 = 15 5U s22 = 56 7U s23 = 78 7: (A-,5-9 &#<#%%2H ,#*%HH -49#'-A#,2H 693--A#,-1 *-,3#,-. : s2 = 14 1; 3 546 = 49 64U (n ; k) lg s2 = (14 ; 3) 1 696 = 18 656: lg s2 = 1 696U 5## (A-,5-9 118 i s2i 1 15,5 2 56,7 3 78,7 ',H* %.*#9 ,2992 lg s2i 1,190 1,754 1,895 3 X i=1 ni ; 1 4 3 4 (ni ; 1) lg s2i 4,760 5,262 7,580 (ni ; 1) lg s2i = 17 602: ,'+%%+ C % 1 1 1 1 1 C = 1 + 3 2 4 + 3 + 4 ; 11 = 1 124: ,,A-'#9 %?5H*#9# &%A#%-# (?A%. ,''-,'-- 2: 2 = 21 303 124 (18 656 ; 17 602) = 2 049 1 054 = 2 160: .*#9 3 '?5-$# %'-5#. 2-,3#*#5#%-+ 3 2%H &%A-9,'- = 0 05 - A-,52 ,'#3#%#. ,?*( = k ; 1 = 2 &%A#%-# -'-A#,. 'A- 20052 = 5 99. 2 = 2 160 < 5 99, ' %#' ,%%-. *5+ '5%#%-+ %25#. 8-3'#&(. D' &%A-', A' -9#HO+,+ -%49$-+ ,,#-%-- 3A%,'- % ,:'-# '#1 9 ?#'% %# *#' ,%%-. ,A-'', A' -1 2#% ,,#-%-+ +5+#',+ &5-A%(9. ' 4.8 29+ 9#'*9- 3-&#*#%( -&9##%-+ *%. - '. :# 4-&-A#,. #5-A-%(. #(9 9#'*9 6' #5-A-% -&9#+5, 10 &. 52A#%( ,5#*2HO-# #&25''( : 10 X 1 2 x1 = 10 28 s1 = n ; 1 (xi1 ; x1)2 = 0 00084: 1 i=1 '(9 9#'*9 6' :# #5-A-% -&9#+5, 8 &, A' *5 8 X x2 = 10 30 s22 = n 1; 1 (xi2 ; x2)2 = 0 00041: 2 i=1 :% 5- ,A-'', A' ? 9#'* ?#,3#A-H' *-%2H 'A%,' ? #% &%A-9,'- 3-%+' = 0 05. #*358#',+, A' #&25''( -&9##%-. ,3#*#5#%( %95% - (?- %#&-,-9(. () & 2,5-+ 3-9# ,5#*2#', A' %9 %#?1*-9 3#-' %25#2H 8-3'#&2 fH0 : 12 = 22g ( $ $ ) 3'- 5'#%'-%. 8-3'#&( fHa : 12 > 22g ( $ ). 119 (A-,5-9 %?5H*#9(# &%A#%-+ F --'#-+: F = 00 00084 = 2 05: 00041 '?5-$# %'-5#. F -,3#*#5#%-+ (,9. 3-5:#%-#) 3 2%H &%A-9,'- = 0 05 - A-,52 ,'#3#%#. ,?*( 1 = 10 ; 1 = 9 - 2 = 8 ; 1 = 7 %1*-9 -'-A#,2H 'A2 F0059 7 = 3 68. F = 2 05 < 3 68, ' %#' ,%%-+ *5+ '5%#%-+ %25#. 8-3'#&(. 28-9- ,59-, -9#HO+,+ -%49$-+ 'A%,'- 6'-1 9#'* %# *#' ,%%-+ ,A-'', A' '. 9#'* -&9##%-+ 52A<# 3#8. ' 4.9 7233 ,$-58 -,,5#*5 5-+%-# ,': ?'( 3 34#,,-- % 3-&*-'#5%,' '2* ?A-1 9#1%-A#,8 $#1 %#'8 &*. 52A#%( ,5#*2HO-# #&25''( : ': ?'( #&25''-%(. ' 10 5#' ' 15 5#' 3-&% * 10 5#' * 15 5#' * 25 5#' 5-A#,' *#'5#., 135 176 155 (?'(#9(1 & 156 196 160 ,9#%2 *%-9 ?A-9, 165 204 149 <'2 | 180 171 | | 140 #*358+, A' 3-&*-'#5%,' '2* ?A-1, -9#HO-1 &5-A%(. ,': ?'(, 3*A-%+#',+ %95%92 &%2: Xi ! N (aiU i) (i = 1 2 3), 3-A#9 12 = 22 = 32, '#?2#',+ 3#-' 9#'*9 *-,3#,-%%8 %5-& %25#2H 8-3'#&2 fH0 : a1 = a2 = a3g ( 1 ). #% &%A-9,'- = 0 05. () 85,% 2,5-H 3-9#, %25#+ - 5'#%'-%+ 8-3'#&( -9#H' -* fH0 : a1 = a2 = a3g | 1 , Ha : a1 = 6 a2 6= a3 | 1 . (A-,5-9 ,398'#5%(# #5-A-%(, %#?1*-9(# *5+ ,,'5#%-+ '?5-$( *-,3#,-%%8 %5-& : n = n1 + n2 + n3 = 3 + 4 + 5 = 12U 756 = 189U 775 = 155U x1 = 456 = 152U x x 2= 3= 3 4 5 < x > = 1987 12 = 165 58: 4 3 X X (x2j ; x2)2 = 524U (x1j ; x1)2 = 474U j =1 j =1 120 5 X j =1 (x3j ; x3)2 = 542U (x1; < x >)2 n1 = 553 25U (x3; < x >)2 n3 = 559 68U ni 3 X X i=1 j =1 (xij ; xi)2 = 1540U (x2; < x >)2 n2 = 2193 99U 3 X (xi; < x >)2 ni = 3306 92: i=1 ,'-9 '?5-$2 *-,3#,-%%8 %5-& : ,'A%- 299( R-,5 -,3#,-+ ?5H*#%%# -&9#%A-,'- *' ,'#3#%#. &%A#%-# ,?*( F --'#-+ ': ?'( (9#:*2 82339-) 3306,92 2 1653,45 F = 9 66 ,''A%+ (%2'- 8233) 1540 9 171,11 5%+ -&9#%A-,' 4846,92 11 .*#9 3 '?5-$9 %'-5#. F -,3#*#5#%-+ -'-A#,# &%A#%-# F 1 2 , ,'#','2HO## 2%H &%A-9,'- - A-,59 ,'#3#%#. ,?*( 1 = k ; 1 = 2 2 = n ; k = 9 (,9. 3-5:#%-#). D' &%A#%-# % F00529 = 4 26. F = 9 66 > 4 26, ' %25#+ 8-3'#& '#8#',+, '.#. ,A-'#',+ ,''-,'-A#,- *&%%(9, A' 3-&*-'#5%,' '2* &-,-' ' ,': ?'(. ' 4.10 %#'%(9 ?,5#*%-#9 2,'%5#%, A' 32% ,'2*#%' 2%-#,-'#' +5+H',+ ,52<'#5+9- '#5#-&-%%(1 5#$-. 3 9'#9'-A#,. ,''-,'-#. 4#* 3-5*%. 9'#9'-- %3#A'5 %,3#'( '#5#-&-%%(1 5#$-. - &,55 -1 ,'2*#%'9. ,5# 6'8 % ?(5 3-&#*#% %#'-%-# - 2,'%5#%, A' 80 ,'2*#%' -& 200 3<#%%(1 &5-, ,52<'#5+9- '#5#-&-%%(1 5#$-., m=n = 80=200 = 0 4. :% 5- ,A-'', A' -&*%-# '#5#-&-%%8 %,3#' 5#$-- ,3,?,'5 2#5-A#%-H %'-%8#%' ,'2*#%', ,52<HO-1 '#5#-&-%%(# 5#$-- 3 9'#9'-A#,. ,''-,'-# (2#% &%A-9,'- = 0 05) ? () 85,% 2,5-H, %#?1*-9 3#-' %25#2H 8-3'#&2 fH0 : p = 32g 3'- 5'#%'-%. 8-3'#&( fHa : p > 32g. 5+ 3#- *%%. %25#. 8-3'#&( 3-9#%-9 u--'#-. , 3,'%%#. -'-A#,. ?5,'H. (A-,5-9 %?5H*#9# &%A#%-# -'#-+ p u = m=n p p ;q p0 pn = p0040 32; 0032 68 200 = 2 43: 0 0 '?5-$9 42%$-- K35, (,9. 3-5:#%-#) 3 &*%%92 2%H &%A-9,'- = 0 05 %.*#9 -'-A#,# &%A#%-# u, 2*5#'+HO## 2,5-H Pr(u u). 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X 40,28] 40,32] 40,36] 40,40] 40,44] R,','- mi=n 0,05 0,24 0,40 0,19 0,12 35#%%(# A,','- F (x) 0,05 0,29 0,69 0,88 1,00 6 %'5 : P m =n = 1. i=1 i 137 2. 5+ 3,'#%-+ 8-,'899( A,','#. % ,- 0x '5*(H',+ A,'-A%(# -%'#5(, % :*9 -& %-1 ,'-',+ 3+9285%-, 35O* '8 % A,','- *%%8 A,'-A%8 -%'#5. ,5- A,','- '%#,'- ,##*-%9 A,'-A%(1 -%'#5, ' 352A#%%+ &9%2'+ 5-%-+ ?&2#' ! A,','#.. -,. 5.1 -&?:#% 8-,'899 - 35-8% A,','#.. 3. %A#%-+ 693--A#,. 42%$-- ,3#*#5#%-+ (3-,%( 3,5#*%#. ,'# ,''-,'-A#,8 +* ,3#*#5#%-+ A,','#.. 3-<#9 &%A#%-+ 693--A#,. 42%$-- ,3#*#5#%-+ %5-'-A#,9 -*# : 8 0 00 #,5x 40 24U > > 0 05 #,5- 40,24 < x 40 28U > > > 0 29 #,5- 40,28 < x 40 32U < #,5- 40,32 < x 40 36U F (x) = 0 69 > 0 88 #,5- 40,36 < x 40 40U > > > 1 00 #,5- 40,40 < x 40 44U > : 1 00 #,5x > 40 44. 2 . %A#%-+ 693--A#,. 42%$-- ,3#*#5#%-+ '%#,#%( #1%#. 8%-$# A,'-A%8 -%'#5. 74- 693--A#,. 42%$-- -&?:#% % -,. 5.2. 4. 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LL ee e TT TT # # T #40.24 40.32 40.40 x 5.1 | 8+ + = 40 26 4 + 40 30 19 + 40 3480 32 + 40 38 15 + 40 42 10 = 40 344U 5 X 143 = 0 0018U D^ ZX] = s2x = n1 (xi ; x)2mi = 080 i=1 q sx = 0 0018 = 0 042U 5 X A^ = ns1 3 (xi ; x)3mi = 80 00001 0423 = 0 17U x i=1 5 X E^ = ns1 4 (xi ; x)4mi ; 3 = 80 0 0006 0424 ; 3 = ;0 589U x i=1 Var = sxx 100 % = 400042 344 100 % = 0 104 %: 5. 5+ 3#*-'#5%8 (? -,98 &% ,3#*#5#%-+ (A-,5-9 %A5# ,#*%-# *'-A#,-# <-?- 3#*#5#%-+ ,-99#'-v s u u 6( n ; 1) 6 79 = 0 266 sA = t (n + 1)(n + 3) = 81 83 - 6,$#,, v s u u 24 n ( n ; 2)( n ; 3) 78 77 = 0 515: sE = t (n ; 1)2(n + 3)(n + 4) = 2479 280 83 84 -'#-#9 ,3#*#5#%-+ *-9#' '#,'-. 3 %95%92 &%2 +5+#',+ #%,' %25H ,-99#'-- - 6,$#,,. & 3-#*#%%(1 ,A#' -*%, A' (?A%(# 644-$-#%'( ,-99#'-- A^ - 6,$#,, E^ '5-AH',+ ' %25+ %# ?5## 139 F (x) 6 1.0 % % % 0.8 0.6 0.4 0.2 0 ( ( ( 40.24 40.32 40.40 x - 5.2 | 7 # () A#9 % 2*#%%(# ,#*%-# *'-A#,-# <-?- -1 3#*#5#%-+, A' ,'#','2#' %95%92 &%2 ,3#*#5#%-+. -* 35-8% - 8-,'899( A,','#. 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( x ; 40 344) 1 exp ; 2 0 0422 : f (x) = p 2 0 042 140 J2%$-+ ,3#*#5#%-+ 3#*358#98 %95%8 &% Zx (x0 ; 40 344)2 ! 1 F (x) = p exp ; 0 0036 dx0: 2 0 042 ;1 ,35&2+ 42%$-H K35, a(x), 42%$-H ,3#*#5#%-+ %95%8 &% 9:% &3-,' -*# ! x ; 40 344 1 F (x) = 2 + a 0 042 : 7. #*#9 *#'5%2H 3#2 8-3'#&( ,3#*#5#%-- X (*-9#' '#,'-.) 3 %95%92 &%2 , 39OH -'#-+ ,85,-+ 2. 5+ 6'8 -%'#5( %?5H*#%%(1 &%A#%-. '%9-2#9, '.#. (&-9 -1 #*-%-$1 ,#*%#8 *'-A#,8 '5%#%-+ s: x ui = xi ; s 3-A#9 %-9#%<## &%A#%-# ui 3-9#9 %(9 ;1, %-?5<## %(9 +1. 5## (A-,5-9 #+'%,'- 33*%-+ X, ,3#*#5#%%. 3 %95%92 &%2 , 39#'9- a = 40 344 - = 0 042, A,'-A%(# -%'#5( Zxi;1U xi] 3 4925# pi = Pr (xi;1 < X < xi) = Za(ui) ; a(ui;1)] 8*# a(z) | 42%$-+ K35, Zz 1 exp(;t2=2) dt: a (z) = p 2 ;1 3-9#, #+'%,' '8, A' X (*-9#' '#,'-.) 33*#' 3#(. A,'-A%(. -%'#5 Z;1U 40 28], % p1 = Pr(;1 < X < 40 28) = " ! !# 1 40 28 ; 40 344 ;1 ; 40 344 = 2 a ;a = 0 042 0 042 = 12 Za(1) ; a(1 52)] = 12 Z1 ; 0 87] = 0 065: %58-A% p2 = Pr(40 28 < X < 40 32) = " ! !# 1 40 32 ; 40 344 40 28 ; 40 344 = a ;a = 2 0 042 0 042 = 12 Za(;0 57) ; a(;1 52)] = 12 Z;0 4314 + 0 8714] = 0 220 - ' *5## *5+ p3, p4, p5. 141 ,5# 6'8 (A-,5-9 '##'-A#,-# A,''( %95%8 &% ,3#*#5#%-+ n = npi - %?5H*#9# &%A#%-# -'#-+ 2 5 X npi )2 : 2 = (mi ; npi i=1 '#9 3 '?5-$9 %'-5#. 2-,3#*#5#%-+ 3 2%H &%A-9,'- = 0 05 - A-,52 ,'#3#%#. ,?*( = k ; r ; 1 (k = 5 | A-,5 -%'#5, r = 2 | A-,5 39#' 3#*358#98 &% ,3#*#5#%-+ X) %.*#9 (,9. 3-5:#%-#) -'-A#,# &%A#%-# 2005 . ,5- 2 2005 , ' 3-9#9, A' %#' ,%%-. *5+ '5%#%-+ 8-3'#&( ,3#*#5#%-- *-9#' '#,'-. 3 %95%92 &%2. 3'-%9 ,52A#, '. #. #,5- 2 > 2005 , ,A-'#',+, A' 8-3'#& ,3#*#5#%-+ *-9#' '#,'-. 3 %95%92 &%2 %# ,85,2#',+ , 6,3#-9#%'5%(9- *%%(9-. (A-,5#%-+, %#?1*-9(# *5+ 3#*#5#%-+ %?5H*#98 &%A#%-+ 2 (?A%. ,''-,'-- 2, ,#*#%( '?5-$2, '. ?&%A#%: Q2i = (mi ; npi )2 - R2i = (mi ; npi )2=(npi ). 2 . - 3,'#%-- '?5-$( %-9#%<## &%A#%-# ,'%*'-&%%. 3##9#%%. (40 24 ; 40 344)=0 042 = ;2 48 &9#%#% % ;1, %-?5<## &%A#%-# (40 44 ; 40 344)=0 042 = +2 29 &9#%#% % 1. D' &9#% 3-&#*#% *5+ '8, A'?( ,299 '##'-A#,-1 A,'' npi ?(5 % ?T#92 (?-. %'#5( R,{ 9-%i %?5H*#%%(1 ''( %(# -%'#- pi &%A#%-. X mi 5( uiU ui+1 1 40,24 { 40,28 4 ;1U ;1 52 0,064 2 40,28 { 40,32 19 ;1 52U ;0 57 0,220 3 40,32 { 40,36 32 ;0 57U 0 38 0,364 4 40,36 { 40,40 15 0 38U 1 33 0,260 5 40,40 { 40,44 10 1 33U 1 0,092 299 80 1,000 ', #&25''# (A-,5#%-. 352A-9 5 X 2 = R2i = 3 17: npi Q2i R2i 5,1 1,21 0,24 17,6 1,96 0,11 29,1 8,41 0,29 20,8 33,64 1,62 7,4 6,76 0,91 80,0 i=1 .*#9 '#3# 3 '?5-$# %'-5#. 2-,3#*#5#%-+ 3 2%H &%A-9,' = 0 05 - A-,52 ,'#3#%#. ,?*( = k ; r ; 1 = 5 ; 2 ; 1 = 2 -'-A#,# &%A#%-# 20052 = 5 99 (,9. 3-5:#%-#). 2 = 3 17 < 5 99, ' %#' ,%%-. *5+ '5%#%-+ 8-3'#&( %95%9 &%# ,3#*#5#%-+ *-9#' '#,'-.. 5## 3#-9 8-3'#&2 ,3#*#5#%-+ *-9#' '#,'-. 3 %95%92 &%2 , 39OH --'#-+ 598. 6'. $#5H *5+ :*8 &%A#%-+ xi %.*#9 9*25 &%,'- 9#:*2 693--A#,. - '##'-A#,. 42%$-+9142 ,3#*#5#%-. jF (x) ; F (x)j - (A-,5-9 %?5H*#9# &%A#%-# (?A%. ,''-,'-- 598: p p = D n = max x jF (x) ; F (x)j n: ?5H*#9# &%A#%-# ,''-,'-- 598 ,%-9 , -'-A#,-9 &%A#%-#9, 3#*#5+#9(9 3 2%H &%A-9,'- = 0 05 (,9. 3-5:#%-#). ,5- 005, ' ,A-'#',+, A' 8-3'#& %95%8 ,3#*#5#%-+ -,,5#*2#9. ,52A.%. #5-A-%( ,85,2#',+ , 6,3#-9#%'5%(9- *%%(9-, #,5:# > 005, | %# ,85,2#',+ , 6,3#-9#%'5%(9- *%%(9-. 5&2+, --'#-#9 ,85,-+ 598, 3#-9 8-3'#&2 %95%8 ,3#*#5#%-+ *-9#' '#,'-.. ,# ,398'#5%(#p ,A#'(, '(# %#?1*-9( *5+ (A-,5#%-+ (?A%. ,''-,'-- = D n, ,#*#9 '?5-$2, '. ?&%A#% m:: | %35#%%(# 693--A#,-# A,''(U p: | %35#%%(# #+'%,'-U D(x) = jF (x) ; F (x)j | '#2O-. 9*25 &%,'-. i 1 2 3 4 5 %'#5( R,'&%A#%-. '( mi m:: X 40,24U 40,28 4 4 40,28U 40,32 19 23 40,32U 40,36 32 55 40,36U 40,40 15 70 40,40U 40,44 10 80 299 80 p: F (x) F (x) D(x) 0,064 0,220 0,364 0,260 0,092 0,050 0,288 0,688 0,875 1,000 0,064 0,284 0,648 0,908 1,000 0,014 0,004 0,040 0,033 0,000 ,9'-+ 3,5#*%-. ,'5?#$ '?5-$(, &9#A#9, A' %-?5<-. 9*25 &%,'- 9#:*2 693--A#,. - '##'-A#,. 42%$-+9- ,3#*#5#%-+ ,,'5+#' D = max x jF (x) ; F (x)j = 0 04: (A-,5-9 %?5H*#9# &%A#%-# (?A%. ,''-,'-- 598 : p p = D n = 0 04 80 = 0 36: -9#9 2#% &%A-9,'- = 0 05. '?5-$9 %'-5#. -,3#*#5#%-+ 598 (,9. 3-5:#%-#) 3 2%H &%A-9,'- = 0 05 %1*-9 #5-A-%2 | -'-A#,# &%A#%-# 005 = 1 358. = 0 36 < 1 358, ' %#' ,%%-+ *5+ '5%#%-+ 8-3'#&( %95%9 &%# ,3#*#5#%-+ *-9#' '#,'-.. 5+ 3,'#%-+ %95%. -. -& ,##*-% A,'-A%(1 -%'#5 ,,'%-9 3#3#%*-25+( (,'. pi =h (pi | #+'%,' 33*%-+ X A,'-A%(. -%'#5, h | *5-% -%'#5). -,. 5.3 %$( 6'-1 3#3#%*-25+ '9#A#%( 2:9-. 52A#%%(# 'A- ,#*-%#%( 35%. -.. %#%-# 8-,'899( - %95%. -. %85+*% 3&(#', A' %95%+ -+ 1< ,85:-#' 8-,'8992 '%,-'#5%(1 A,''. 8. .*#9 -%'#5%(# $#%- 39#' %95%8 &% ,3#*#5#%-+. 143 mi=nh 6 7.5 x = 40:344 5.0 2.5 0 40.24 40.32 x 40.40 x - 5.3 | 8+ 5+ %1:*#%-+ *#-'#5%8 -%'#5, '(. ,*#:-' 9'#9'-A#,# :-*%-# *-9#' '#,'-+ (X), %.*#9 3 '?5-$9 %'-5#. ,3#*#5#%-+ 'H*#%' 3 &*%%. *#-'#5%. #+'%,'- P = 1; = 0 95 - A-,52 ,'#3#%#. ,?*( = n ; 1 = 80 ; 1 = 79 %'-5 t=2 = t002579 = 1 99. (A-,5-9 3#*#5%2H 38#<%,' -%'#5%8 $#%-%-+ " = t=2 psn = 1 99 0p 042 = 0 009: 80 ,%%-- 4925( x;" < a < x+" %.*#9, A' -,9(. *#-'#5%(. -%'#5 *5+ 9'#9'-A#,8 :-*%-+ #% 40 344 ; 0 009 < a < 40 344 + 0 009: ', 352A#9 *#-'#5%(. -%'#5 *5+ 9'#9'-A#,8 :-*%-+ 40 335 < a < 40 353: 9(,5 352A#%%8 #&25'': #,5- ?2*#' 3-&#*#% *,''A% ?5<# A-,5 (? 3 80 ,#5#%-. '#,'-., ' 95 % -& %-1 *#-'#5%(. -%'#5 %#' 9'#9'-A#,# :-*%-# *-9#' '#,'-+ - '5 5 % ,52A# 9'#9'-A#,# :-*%-# 9:#' (.'- & 8%-$( *#-'#5%8 -%'#5. 5+ %1:*#%-+ *#-'#5%8 -%'#5, ,*#:O#8 %#-&#,'%# ,#*%## *'-A#,# '5%#%-# , &*%%. #+'%,'H P = 1 ; = 0 95, %.*#9 3 '?5-$9 %'-5#. ,3#*#5#%-+ 'H*#%' 3 *#-'#5%. #+'%,'P = 1 ; = 0 95 - A-,52 ,'#3#%#. ,?*( = n ; 1 = 80 ; 1 = 79 * A-,5: 1 = 0 87 - 2 = 1 18. 144 ,%%-- 4925( 1s < < 2s %.*#9, A' -,9(. *#-'#5%(. -%'#5 #% 0 87 0 042 < < 1 18 0 042: ', 352A#9 *#-'#5%(. -%'#5 *5+ ,#*%#8 *'-A#,8 '5%#%-+ 0 037 < < 0 050: 52A#%%(. #&25'' &%A#', A' #,5- ?2*#' 3-&#*#% *,''A% ?5<# A-,5 (? 3 80 ,#5#%-. '#,'-., ' 95 % -& %-1 *#-'#5%(. -%'#5 %#' ,#*%## *'-A#,# '5%#%-# - '5 5 % ,#*%## *'-A#,# '5%#%-# 9:#' (.'- & 8%-$( *#-'#5%8 -%'#5. 5.7. ' 5.1 3#(1 *21 ,'5?$1 '?5-$( 3-#*#%( *%%(# ? '&1 33'2( & 10000 A, ?'(. ?O## A-,5 ?,5#*%%(1 6&#935+ 33'2( n = 757U 3- 6'9 %?5H*5,+ 0 427 + 1 235 + 2 72 + 3 21 + 4 1 + 5 1 = 451 '&: R-,5 '& k 5-A#,' ,52A# , k '&9- nk 0 427 1 235 2 72 3 21 4 1 5 1 -%+ 2#% &%A-9,'- = 0 01, 3#-' 8-3'#&2 '9, A' A-,5 '& -9##' ,3#*#5#%-# 2,,% : k pk = PrZX = k] = k! exp(;) k = 0 1 2 : : : : () A#,'# $#%- ^ 39#' -,35&2#9 ,#*%## A-,5 '& : ^ = 451=757 0 6: '?5-$# ,3#*#5#%-+ 2,,% , = 0 6 %1*-9 #+'%,'- pk :-*#9# A-,5 ,52A# , k '&9- ('#'-. - A#'#'(. ,'5?$( '?5-$(). 145 R-,5 5-A#,' #+'%,' :-*#9# k 0 6 '& ,52A# , k pk = k! exp(;0 6) A-,5 ,52A# , k '&9- nk k '&9- npk 0 427 0,54881 416 1 235 0,32929 249 2 72 0,01976 75 3 21 0,00296 15 4 1 0,00036 2 5 1 0,00004 0 5+ k = 4 - 5 &%A#%-+ npk < 5, 36'92 ?T#*-%+#9 6'- ,'- , ,'. *5+ k = 3. #&25''# 352A#9 &%A#%-+, 3-#*#%%(# '?5-$# : k 0 1 2 3 9##9 nk 427 235 72 23 npk 416 249 75 17 (nk ; npk )2=npk 0,291 0,787 0,120 2,118 3 (n ; np )2 X k k = 3 316: np k k=0 3 (?# $#%-5,+ *-% 39#' , ' l = 1, A-,5 ,'#3#%#. ,?*( % 4 { 1 { 1 = 2. '?5-$# %'-5#. 2-,3#*#5#%-+ -,% %1*-9 20992 = 9,21. ', -9##9 2 < 20992. 5#*'#5%, 3- *%%9 2%# &%A-9,'8-3'#& ,3#*#5#%-- A-,5 '& 3 &%2 2,,% 3-%-9#',+. 2 = ' 5.2 9#H',+ *%%(# ,3#*#5#%-- '5O-%( X 12000 ?? ( ) : } -%'#5 1 2 3 4 5 6 7 8 A,'' 32 103 239 624 1187 1650 1883 1930 } -%'#5 9 10 11 12 13 14 15 16 A,'' 1638 1130 737 427 221 110 57 32 *#, 3#(. -%'#5 | &%A#%-+ X, 9#%<-# 7,00 , '. | 7,00{7,25, '#'-. | 7,25{7,50 - '.*. #?2#',+ 3#-', ,85,2#',+ 5- '5O-% ?? (?# , 3#*35:#%-#9, A' 6'' 3-&% 8#%#5%. ,23%,'- ,3#*#5#% 3 %95%92 &%2 ? () -,1*%(1 *%%(1 3-9#2 ',2','2H' ,#*#%-+ 39#'1 %95%8 ,3#*#5#%-+, ' A#,'# 39#' X -,35&2#9 X~ , &%A#%-# 146 '8 *%%9 ,52A# % 8,523, & 8#%#5%2H *-,3#,-H | #E %#,9#O#%%2H $#%2 n 2 = 0 3798 s2 = n ; 1 '2* s = 0,6163. 1*-9 #5-A-%( npi *5+ :*8 -& -%'#5. -9 ?&9, 352A-9 ,5#*2HO-. +* '##'-A#,-1 A,'' : 67,6U 132,2U 309,8U 617,3U 1045,7U 1505,8U 1842,3U 1919,9U 1697,9U 1277,3U 817,0U 444,2U 205,3U 80,7U 27,0U 10,0. :#9, 3-&*-',+ ,A#' % 3-9## '8 -%'#5 : np2 = 1200 Pr(7 00 X 7 25) = " ! !# 1 7 25 ; 8 712 7 00 ; 8 712 ; a 0 6163 = 132 2: = 1200 2 a 0 6163 6'92 20 = 6)2 + (103 ; 132 2)2 + + (57 ; 27 0)2 + (32 ; 10 0)2 = (32 ;6767 6 132 2 27 0 10 0 A' *#' 20 = 196 5: * 39#' 8#%#5%8 ,3#*#5#%-+ ?(5- %.*#%( 3 (?#, ' s = 2, A-,5 ,'#3#%#. ,?*( k { s { 1 = 16 { 2 { 1 = 13. - = 0,01 8%-$#. -'-A#,. ?5,'- +5+#',+ 2 = 27,688. 20 = 196 5 > 27 688, ' 8-3'#& '9, A' ,52A.%+ #5-A-% X ('5O-% ?? (?#) 8#%#5%. ,3#*#5#% 3 %95%92 &%2, %# 3*'#*-5,. ' 5.3 '#A#%-# 100 *%#. 4-,-5, 5-A#,' -. *3*%. ,#'- %#'9 .%# 8*. 52A#%( ,5#*2HO-# A-,5(# *%%(# : R-,5 -. ( X) 0 1 2 3 4 5 R,''( mi 8 28 31 18 9 6 P n = i mi = 100 #?2#',+ 3#-' 8-3'#&2 '9, A' ,3#*#5#%-# A-,5 -. *3*%. ,#'- 8* 3*A-%+#',+ &%2 2,,%. #% &%A-9,'- 3-%+' = 0 05. () & 2,5-+ ,5#*2#', A' %#?1*-9 3#-' %25#2H 8-3'#&2 fH0: F (x)g | # x i X F (x ) = exp(;) i=1 i! 147 , 39#'9 X ^ = x = n1 mixi = i 8 0 + 28 1 + 31 2 + 18 3 + 9 4 + 6 5 = 2 1: = 100 (A-,5-9 '##'-A#,-# #+'%,'- pi 3+5#%-+ % xi -. '#A#%-# n *%#. 3 4925# 2,,% : pi = Pn (xi) = x1 ! xi exp(;)U xi = 0 1 2 3 4 5: i 5%#.<-# (A-,5#%-+ ,*-9 '?5-$2. R-,5 -. xi mi pi npi (mi ; npi)2 (mi ; npi)2=npi 0 8 0,122 12,2 17,64 1,45 1 28 0,257 25,7 5,29 0,21 2 31 0,270 27,0 16,00 0,59 3 18 0,189 18,9 0,81 0,04 4 9 0,099 9,9 0,81 0,08 5 6 0,063 6,3 0,03 0,01 299(: 100 1,000 100,0 40,58 2,38 #&25''# (A-,5#%-. %1*-9 2 = 2 38. '?5-$# %'-5#. 2-,3#*#5#%-+ 3 &*%%92 2%H &%A-9,' = 0 05 - A-,52 ,'#3#%#. ,?*( = k ; r ; 1 = 6 ; 1 ; 1 = 4 %.*#9 -'-A#,# &%A#%-# 20054 = 9 488. 2 = 2 38 < 9 488, ' %#' ,%%-. *5+ '5%#%-+ 8-3'#&( '9, A' &% ,3#*#5#%-+ A-,5 -. *3*%. ,#'- +5+#',+ &%9 2,,% , 39#'9 = 2 1. ' 5.4 #&25''( -,,5#*%-+ 3A%,'- % ,:'-# (,52A.%+ #5-A-% X) 200 ?&$ ?#'% 3#*,'5#%( -*# ,8233-%%8 ,''-,'-A#,8 +*. i 1 2 3 4 5 6 %'#5( 3A%,'-, !/2 190{200 200{210 210{220 220{230 230{240 240{250 R,''( mi 10 26 56 64 30 14 #?2#',+ 3#-' %25#2H 8-3'#&2 %95%9 &%# ,3#*#5#%-+ 3A%,'- % ,:'-#. #% &%A-9,'- 3-%+' = 0 05. () & 2,5-+ ,5#*2#', A' 'A%(# 39#'( 8-3'#'-A#,8 %95%8 &% %9 %#-&#,'%(, 36'92 %25#2H 8-3'#&2 ,5#,% 9:% ,4925-' 148 ,5#*2HO-9 ?&9: fH0: F (x) 2 ^g +5+#',+ 42%$-#. %95%8 ,3#*#5#%-+ , 39#'9MZX] = â - DZX] = ^ 2 = s2: 5+ 3#- 6'. %25#. 8-3'#&( 3#*#5-9 &%A#%-# xi ,##*-% -%'#5 - %.*#9 'A#A%(# $#%- 9'#9'-A#,8 :-*%-+ - ,#*%#8 *'-A#,8 '5%#%-+ %95% ,3#*#5#%%. ,52A.%. #5-A-%( 3 49259 (n = 200): 6 1 X a^ = x = 200 xi mi = i=1 + 225 64 + 235 30 + 245 14 = 221 !/2U = 195 10 + 205 26 + 215 56200 6 1 X ^ 2 = s2 = 200 (xi ; x)2 mi = i=1 1 Z(;26)2 10 + (;16)2 26 + (;6)2 56 + 42 64 + 142 30 + 242 14] = 152U = 200 p ^ = s = 152 = 12 33 !/2: (A-,5-9 '##'-A#,-# #+'%,'- pi 33*%-+ ,52A.%. #5-A-%( X A,'-A%(# -%'#5( Zxi;1U xiZ 3 4925# pi = Pr (xi;1 X < xi) = a(ui) ; a(ui;1)U 8*# i = 1 2 : : : k Z ui ;x 1 x i ui = s U a(ui) = p 0 exp(;t2=2) dt: 2 5%#.<-# (A-,5#%-+, %#?1*-9(# *5+ 3#*#5#%-+ %?5H*#98 &%A#%-+ (?A%. ,''-,'-- 2, ,#*#%( '?5-$#, '. ?&%A#% Qi = (mi ;npi )2 - Ri = (mi ; npi)2=npi . %'#5( 9-%%(# i -&9#%#%-+ mi -%'#5( pi npi Qi Ri 1 190{200 10 ;1U {1,70 0,045 9 1,00 0,11 2 200{210 26 {1,70U {0,89 0,142 28,4 5,76 0,20 3 210{220 56 {0,89U {0,08 0,281 56,2 0,04 0,00 4 220{230 64 {0,08U +0,73 0,299 59,8 17,64 0,29 5 230{240 30 +0,73U +1,54 0,171 34,2 17,64 0,52 6 240{250 14 +1,54U 1 0,062 12,4 2,56 0,21 299(: 200 1,000 200,0 #&25''# (A-,5#%-. %1*-9: 2 = 1 33. 2 . ,52A.%+ #5-A-% X, ,3#*#5#%%+ 3 %95%92 &%2, 3#*#5#% % (;1U 1), ' %-9#%<## &%A#%-# ,'%*'-&%%. 3##9#%%. (190 ; 221)=12 33 = ;2 51 &9#%#% % ;1, %-?5<## &%A#%-# (250 ; 221)=12 33 = 2 35 &9#%#% % +1. 149 '?5-$# %'-5#. 2-,3#*#5#%-+ 3 &*%%92 2%H &%A-9,' = 0 05 - A-,52 ,'#3#%#. ,?*( = k ; r ; 1 = 6 ; 2 ; 1 = 3 %.*#9 -'-A#,# &%A#%-# 20053 = 7 815. 2 = 1 33 < 7 815, ' %#' ,%%-. *5+ '5%#%-+ %25#. 8-3'#&( %95%9 &%# ,3#*#5#%-+ 3#*#5 3A%,'- % ,:'-# , 39#'9- = 221 - 2 = 152. ' 5.5 - 39O- -&9#-'#5%8 3-? ?(5 3#*#% 200 -&9##%-. &*%%8 ,,'+%-+. 52A.%+ 38#<%,' -&9##%-. &3-,% 9-9#'1. #.,'-'#5%+ , ?(5 &*#5#% % 9 39#:2', #&25''( (,52A.%+ 38#<%,' -&9##%-.) ,#*#%( '?5-$2 (*?%(# &%A#%-+ mi %#. 3+-5-, -&-& '8, A' &%A#%-+, 33<-# % 8%-$2 -%'#5, 3-%+' &3-,(' 3%2 *%92, ' - *2892 -%'#52). 9# -%'#5 %'#5, R,'' mi 1 9#%<# {15 6 2 ' {15 * {10 11,5 3 ' {10 * {5 15,5 4 ' {5 * 0 22 5 ' 0 * 5 47,5 6 ' 5 * 10 42 7 ' 10 * 15 28 8 ' 15 * 20 17 9 ?5<# 20 10,5 #?2#',+ 3#-' 8-3'#&2 H0 '9, A' ,52A.%+ 38#<%,' -&9##%-+ X ,3#*#5#% %95%. () #?1*-9(# *#.,'-+ (35%-9 ,5#*2HO#9 3+*# : ) ,,9'-9 8-3'#'-A#,-. (%95%(.) &% ,3#*#5#%-+U ?) 3 &*%%. (?# 352A-9 %-?5## 3*3*?%(# &%A#%-+ 39#' ,3#*#5#%-+U ) 3,'-9 -'#-. 2, '(. -,35&2#9 *5+ 3#- 8-3'#&( %95%,'- ,52A.%. 38#<%,'- -&9##%-+ X. 7-3'#'-A#,# ,3#*#5#%-# -9##' 35'%,' 2! 1 ( x ; a ) fX (xU a ) = p 2 exp ; 2 2 2 '+ ,*#:-' * %#-&#,'%(1 39#' a - . ,'-9 ,398'#5%2H '?5-$2 ,''-,'-A#,8 ,3#*#5#%-# ,52A.%. 38#<%,'- -&9##%-.. 5+ 6'8 3 -9#HO#.,+ (?# -& n = 200 -&9##%-. (A-,5-9 $#%- *5+ a - , A' *#' n X a ) x = n1 xi = 4 60 i=1 150 n X 2 ) 2 = n1 (xi ; x)2 = 95 2 2: i=1 -9 ?&9, 42%$-+ f (x), -,35&2#9+ *5+ 3#-, -9##' -* 2! ( x ; 4 60) 1 f (x) = p 2 95 2 exp ; 2 95 2 : ',H* 3 '?5-$# 42%$-- K35, 9:% %.'- #+'%,'- pi 33*%-+ i-. -%'#5. 3-9#, 3- i = 1 - i = 2 ! ; 15 ; 4 6 X ; 4 6 p1 = Pr(X < ;15) = Pr p 95 2 < p 95 2 = = 21 ; a(2 01) = 12 ; 0 4776 = 0 0224 ! X ; 4 6 p2 = Pr(;15 < X < ;10) = Pr ;2 01 < p 95 2 < ;1 50 = = a(;1 50) ; a(;2 01) = a(2 01) ; a(1 50) = 0 4776 ; 0 4332 = 0 0444 - ' *5## (i = 3 : : : 9). #&25''# ,'-9 '?5-$2 *%%(1, %#?1*-9(1 *5+ %1:*#%-+ &%A#%-+ -'#-+ 2 (,9. '?5-$2 ,A#'%(1 *%%(1). i R,'' mi pi npi (mi ; npi )2=npi 1 6 0,0224 4,48 0,52 2 11,5 0,0444 8,88 0,77 3 15,5 0,0967 19,34 0,76 4 22 0,1557 31,14 2,68 5 47,5 0,1968 39,36 1,68 6 42 0,1928 38,56 0,31 7 28 0,1489 29,78 0,11 8 17 0,0852 17,04 0,00 9 10,5 0,0571 11,42 0,07 2 299 200,0 1,0000 200,00 = 6 90 -9 ?&9, (A-,5#%-# *#' 2 = 6 90. ,52 3-&*-',+ $#% *21 39#', ' -9##9 k = 9 ; 2 ; 1 = 6 ,'#3#%#. ,?*(. -9#9, A' 2#% &%A-9,'- ,,'5+#' = 0 05, '8* -& '?5-$( %'-5#. 2-,3#*#5#%-+ (,9. 3-5:#%-#) 352A#9 20056 = 12 6. (A-,5#%%# &%A#%-# 2--'#-+ &5, 9#%<# %'-5+ 20056, ' 8-3'#& %95%9 ,3#*#5#%-- ,52A.%. 38#<%,'- -&9##%-+ 3-? %# 3'-#A-' 9'#-52 %?5H*#%-. % 2%# &%A-9,'- 0,05. ' 5.6 & '#2O#. 3*2$-- '%8 ,'%-'9', %,'#%%8 % ??'2 &*%%. *#'5-, &+'( *# (?- ?T#9 n1 = 150 - n2 = 100. #+ (? 151 3-&#*#% %A5# ,9#%(, '+ | 3,5# *21 A, ?'( ,'%. #&25''( -&9##%-. '5%#%-. %'5-2#98 &9# ( X) ' %9-%5 9-9#'1 3-#*#%( '?5-$#. %'#5( R,'' R,'' -&9#%#%-+ X, (?# } 1, m1j (?# } 2, m2j {15U {10 10 | {10U {5 27 7 {5U 0 43 17 0U 5 38 30 5U 10 23 29 10U 15 8 15 15U 20 1 1 20U 25 | 1 299 : n1 = 150 n2 = 100 #?2#',+ , 39OH --'#-+ 598-9-% (9*#5 2) 3#-' %25#2H 8-3'#&2 '9, A' ,3#*#5#%-# 38#<%,'#. ??'- ,'%-'9' '#A#%-# -,,5#*2#98 39#:2' #9#%- 3-,(#',+ *%. - '. :# 42%$-#. ,3#*#5#%-+. -%+' 2#% &%A-9,'- = 0 05. () 85,% 2,5-H 3-9#, %#?1*-9 3#-' %25#2H 8-3'#&2 fH0 : F1(x) = F2(x)g (# ! - ). (A-,5#%-# (?A%. ,''-,'-- D = max x jF1 (x) ; F2(x)j 3-#*#% '?5-$#, '. m1j - m2j | A,''(, n1j - n2j | %35#%%(# A,''(, F1(x) = n1(x)=n1 - F2(x) = n2(x)=n2. xi+1 m1i m2i n1(x) n2(x) F1(x) F2(x) jF1(x) ; F2(x)j {10 10 | 10 | 0,067 0,000 0,067 {5 27 7 37 7 0,247 0,070 0,177 0 43 17 80 24 0,533 0,240 0,293 5 38 30 118 54 0,787 0,540 0,247 10 23 29 141 83 0,940 0,830 0,110 15 8 15 149 98 0,993 0,980 0,013 20 1 1 150 99 1,000 0,990 0,010 25 | 1 150 100 1,000 1,000 0,000 %5-&-2+ 3,5#*%-. ,'5?#$ *%%. '?5-$(, &9#A#9, A' %-?5<-. 9*25 &%,'- 9#:*2 693--A#,-9- 42%$-+9- ,3#*#5#%-+ F1(x) - F2(x) #% D = max x jF1 (x) ; F2 (x)j = 0 293. 150 100 = 60 n = nn1+ nn2 = 150 + 100 1 2 ' %?5H*#9# &%A#%-# (?A%. ,''-,'-- p p = D n = 0 293 60 = 2 270: 152 '?5-$# %'-5#. -,3#*#5#%-+ 598-9-% *5+ &*%%8 2%+ &%A-9,'- = 0 05 %.*#9 -'-A#,# &%A#%-# 005 = 1 358. = 2 270 > 1 358, %25#2H 8-3'#&2 ,5#*2#' '5%-'. -9 ?&9, %#5&+ 2'#:*', A' 38#<%,'- ??'- '#A#%-# -,,5#*2#98 -%'#5 #9#%- 3-,(H',+ *%. - '. :# 42%$-#. ,3#*#5#%-+. 28-9- ,59-, 352A#%%(. #&25'' 8-' '9, A' 3$#,, -&8'5#%-+ *#'5#. % ,'%#-'9'# %# +5+#',+ 2,'.A-(9 #9#%-. ' 5.7 935#'2HO-# -&*#5-+ *%8 %-9#%%-+ 3,'23H' , '#1 3#*3-+'-.: A, B - C . #&25''( 3#- -&*#5-. 3-#*#%( ,5#*2HO#. '?5-$# (*5+ 3,'O- A, B - C ) : #&25''( 3#- A B C ,#8 7*%(# 29 38 53 120 #8*%(# 1 2 7 10 ,#8 30 40 60 130 -%+ 2#% &%A-9,'- = 0 10, (+,%-', 9:% 5- ,A-'', A' A#,' -&*#5-. %# &-,-' ' 3,'O- ? () #-9 8-3'#&2 H0 %#&-,-9,'- *21 3-&% : A#,' -&*#5-+ X 9#,' #8 -&8'5#%-+ Y. 5+ 6'8 3 -'#-H 2 -,35&2#9 ,''-,'-2 k X l (n ; n X ij ~ ij )2 2 = () n~ ij : i=1 j =1 *#, nij | A-,5 -,1*, '(1 #5-&5, A##*%# ,?('-# fX = xi - Y = yj g, n~ij | :-*#9(# A,''(. - 2,5--, A' 8-3'#& H0 #% - *5+ ,#1 :-*#9(1 A,'' (35%+#',+ n~ ij 4, ,''-,'- () -9##' ,3#*#5#%-# 2 , (k ; 1)(l ; 1) ,'#3#%+9- ,?*(. %<#9 3-9## 39#'( k - l %( k = 2 - l = 3, ' A' 35%# A-,5 ,'#3#%#. ,?*( ,,'5+#' (k ; 1)(l ; 1) = (2{1)(3{1) = 2. -*+ 49252 () 2*?%92 *5+ (A-,5#%-. -*2, %.*#9 292 2 2 2 2 2 38 53 1 2 7 2 = 130 30 120 + 40 120 + 60 120 + 30 10 + 40 10 + 60 10 ; 1 '2* 2 = 2 546: '?5-$# %'-5#. 2-,3#*#5#%-+ %1*-9 20902 = 4,605. 2 < 20902, ' 3- *%%9 2%# &%A-9,'- ,5#*2#' ,A-'', A' A#,' -&*#5-. %# &-,-' ' 3,'O-. ' 5.8 - -,,5#*%-- 3#*#5 35,'-A%,'- 15 ?&$ 3#*#5#%%8 ,' ,'5352A#%( ,5#*2HO-# #&25''( ( !/2) : 153 3540, 3580, 3570, 3560, 3500, 3610, 3720, 3640, 3600, 3650, 3750, 3590, 3600, 3550, 3770. - *35%-'#5%9 '#1%58-A#,9 3$#,,#, '(. 3#*35:-'#5% 3-#*#' 2#5-A#%-H 3#*#5 35,'-A%,'-, 352A#%( *5+ '#1 :# ?&$ ,5#*2HO-# #&25''( : 3580, 3670, 3680, 3880, 3530, 3680, 3730, 3720, 3670, 3710, 3810, 3660, 3770, 3640, 3670. #-' , 39OH -'#-+ &% 8-3'#&2, A' 3#*#5 35,'-A%,','5- 3- 3#*#%-- *35%-'#5%8 '#1%58-A#,8 3$#,, 2#5-A-5,+. #% &%A-9,'- 3-%+' *%%. &*A# = 0 05. () 4925-2#9 %25#2H 8-3'#&2 fH0 : F (x) = F (y)g | 3#*#5 35,'-A%,'-,,5#*2#98 ,' ,'5- %# 9#%+#',+ 3- 3#*#%-- *35%-'#5%8 '#1%58-A#,8 3$#,,. ?&%A-9 &%9 "+" &,'%-# 3#*#5 35,'-A%,'-, &%9 "{" | 29#%<#%-# 3#*#5 35,'-A%,'-, (&%%# *35%-'#5%(9 3$#,,9. 52A#9 ,5#*2HO2H 3,5#*'#5%,' &%: + { + + + + + + + + + + + + { R-,5 &% "{" ,,'5+#' r = 2. .*#9 3 '?5-$# -'-A#,-1 &%A#%-. A-,5 &% (,9. 3-5:#%-#) 3 &*%%92 2%H &%A-9,'- = 0 05 - ?T#92 (?- n = 15 -'-A#,# &%A#%-# r n = r00515 = 3. r = 2 < 3, ' %25#+ 8-3'#& '5%+#',+. 28-9- ,59-, ,A-'#',+ ,''-,'-A#,- 2,'%5#%%(9, A' *35%-'#5%(. 3$#,, 3-*-' 2#5-A#%-H 3#*#5 35,'-A%,'- *%%8 ,' ,'5-. 5.8. ) * 5.1 52A#%( ,5#*2HO-# *%%(# (,9. '?5-$2). 40,25 40,35 40,45 40,39 40,40 40,42 40,35 40,32 40,37 40,35 40,44 40,30 40,34 40,31 40,35 40,32 40,33 40,41 40,35 40,33 40,38 40,33 40,30 40,33 40,28 40,30 40,40 40,32 40,32 40,42 40,36 40,35 40,29 40,33 40,31 40,36 40,34 40,30 40,33 40,30 40,41 40,40 40,33 40,34 40,30 40,43 40,37 40,34 40,35 40,34 40,34 40,43 40,36 40,34 40,31 40,34 40,28 40,46 40,32 40,31 40,31 40,36 40,34 40,34 40,29 40,39 40,39 40,37 40,38 40,36 40,37 40,41 40,27 40,38 40,37 40,36 40,35 40,32 40,37 40,36 39OH --'#-+ 598 3#-' 8-3'#&2 '9, A' 3-#*#%%+ (? -&5#A#% -& 8#%#5%. ,23%,'-, %9#% ,3#*#5#%%. % -%'#5# (40 238U 40 462). 154 5.2 R,(, (,'5#%%(# -'-%1 A,(1 98&-%, 3&(H' ,52A.%# #9+. #*58#',+ 8-3'#&, A' 3&%-+ 6'-1 A, -'-%1 ?5<8 A-,5 98&-% ,3#*#5#%( %9#% -%'#5# (0U 12). ?5H*#%-+ 500 -'-% 500 98&-% *5- ,5#*2HO2H (?2 (#, -%'#5 (0U 12) &?-' % 12 A,(1 -%'#5): C 0 1 2 3 4 5 6 7 8 9 10 11 C / 41 34 54 39 49 45 41 33 37 41 47 39 85,2H',+ 5- 6'- *%%(# , 8-3'#&. %9#%,'- ? (?' = 0 10 = 0 05. 5.3 -:# 3-*-',+ #9+ ( ,#2%*1) #<#%-+ %'5%(1 &*A *-%%*$'H 2AO-9-,+ * - 3,5# ,3#$-5%(1 23:%#%-. 3 2,'%92 ,A#'2. > % 87 61 98 90 74 83 72 81 75 83 85 $ % 50 45 79 88 65 52 79 84 61 52 85 :% 5- ,A-'', A' 6'- 23:%#%-+ 252A<-5- ,3,?%,' 2AO-1,+ #<#%-H &*A ? -%+' = 0 10. 5.4 - -&9##%-- 3-&*-'#5%,'- *21 8#8' 352A#%( ,5#*2HO-# #&25''( ( ! #O#,' & A, ?'(): } &9# 1 2 3 4 5 8#8' A 14,1 10,1 14,7 13,7 14,0 8#8' B 14,0 14,5 13,7 12,7 14,1 :% 5- ,A-'', A' 3-&*-'#5%,'- 8#8' A - B *-%(, 3#*35:#%--, A' ?# (?- 352A#%( -& %95% ,3#*#5#%%(1 8#%#5%(1 ,23%,'#. ? -%+' = 0 05. 5.5 8* % 3('%(. .% 235 537 %#?5<-1 9#'#-'. ,+ '#-'-+ .% ?(5 &*#5#% % 576 2A,' 35O*H 3 0,25 2 :*(.. -:# 3-#*#%( A-,5 2A,' nk , % '(# 235 k 9#'#-' : k 0 1 2 3 4 5 - ?5<# nk 229 211 93 35 7 1 85,2H',+ 5- 6'- *%%(# , 8-3'#&. '9, A' A-,5 9#'#-', 23<-1 % :*(. -1 2A,', -9##' ,3#*#5#%-# 2,,% ? -%+' = 0 10. 5.6 #*358#',+, A' *-% -& *21 3-?, 3#*#5+HO-1 ,,' '9?-155 5+, -9##' ,-,'#9'-A#,2H <-?2 (&(<#%-#). 5+ 3#- 6'8 3#*35:#%-+ 3#*#5-5- ,,' 10 '9?-5#., 3-A#9 ,,' :*8 -& %-1 4-,-5, *%#9#%% *29+ 3-?9-. #&25''# 352A#%( ,5#*2HO-# *%%(# : v1, / 70 85 63 54 65 80 75 95 52 55 v2, / 72 86 62 55 63 80 78 90 53 57 &5+H' 5- 6'- *%%(# 2'#:*', A' '. 3-? *#.,'-'#5% *#' &(<#%%(# &%A#%-+ ,,'- ? -%+' = 0 10. 5.7 52A#%( ,5#*2HO-# *# 8233( *%%(1 (,9. '?5-$(). 40,25 40,35 40,45 40,39 40,40 40,42 40,35 40,32 40,37 40,35 40,44 40,30 40,34 40,31 40,35 40,32 40,33 40,41 40,35 40,33 40,38 40,33 40,30 40,33 40,28 40,30 40,40 40,32 40,32 40,42 40,36 40,35 40,29 40,33 40,31 40,36 40,34 40,30 40,33 40,30 40,41 40,40 40,33 40,34 40,30 40,43 40,37 40,34 40,35 40,34 40,34 40,43 40,36 40,34 40,31 40,34 40,28 40,46 40,32 40,31 40,31 40,36 40,34 40,34 40,29 40,39 40,39 40,37 40,38 40,36 40,37 40,41 40,27 40,38 40,37 40,36 40,35 40,32 40,37 40,36 39OH -'#-+ 598-9-% 3#-' 8-3'#&2 '9, A' ?# 6'- 3-#*#%%(# (?- -&5#A#%( -& *%. - '. :# 8#%#5%. ,23%,'-. 5.8 ,3('(5, A2,'-'#5%,' 40 3-#9%-. %%(# 3-#*#%( '?5-$#, '. 3#. ,'# *%( -%'#5( A2,'-'#5%,'- 9-5'1, '. | ,#*%-# 'A- 6'-1 -%'#5 f, '#'#. | A-,5 3-#9%- ni, A2,'-'#5%,' '(1 &5, 6'9 -%'#5#. %'#5 25{75 75{125 125{175 175{225 225{275 f 50 100 150 200 250 ni 0 0 1 5 9 %'#5 275{325 325{375 375{425 425{475 475{525 f 300 350 400 450 500 ni 6 8 6 2 2 %'#5 525{575 575{625 625{675 675{725 725{775 f 550 600 650 700 750 ni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i '*#5#%-., ,*#:O-1 3 ni ,%(1 +%(1 <-, 2&%( '?5-$#. i 4 5 6 7 8 9 10 11 12 ni 1 3 5 8 13 14 15 15 21 i 13 14 15 16 17 18 19 20 21 ni 18 17 16 9 6 3 2 2 1 .'- ,#*%## &%A#%-# A-,5 ,%(1 +%(1 <- *%9 '*#5#%--. -%+ %.*#%%# &%A#%-# & 39#' ,3#*#5#%-+ 2,,%, 3#-' , 39OH 2--'#-+ 8-3'#&2 '9, A' (? ,85,2#',+ , ,3#*#5#%-#9 2,,%. 5.12 '#:*#',+, A' #&25'' *#.,'-+ 5#,' &-,-' ' ,3,? #8 3-9#%#%-+. #-' 6' 2'#:*#%-# 3- = 0 05. 3 ,5#*2HO-9 *%%(9 #&25'' 3,? 1 3,? 2 3,? 3 #?583-+'%(. 11 17 16 583-+'%(. 20 23 19 5.13 935-'2* 5#?%-. 3#*#5+5, *29+ 5?%'9-. #(. 5?%' 3 10 %?5H*#%-+9 352A-5 ,#*%## &%A#%-# 935-'2*( x1 = 81 , '. 5?%' 3 15 %?5H*#%-+9 352A-5 x1 = 84 . 3#*35:#%--, A' *-,3#,-- (35%#%%(1 -&9##%-. -&#,'%( - %( 12 = 64 2 - 12 = 64 2 *5+ 3#8 - '8 5?%' ,'#','#%%, %.'99 %{. *#-'#5%(. -%'#5 *5+ &%,'- ,#*%-1 X 1 - X 2. :% 5- ,A-'', A' #&25''( 5?%' *#.,'-'#5% &5-AH',+ ? 157 5.14 D,3#-9#%'' 3- n = 4040 ?,%-+1 9%#'( 352A-5 m = 2048 (3*#%-. 8#?. 9#,'-9 5- 6' , 8-3'#&. '9, A' ,2O#,'2#' 3,'+%%+ #+'%,' p = 0,5 (3*#%-+ 8#? ? 5.15 #*358#',+, A' 3-9#%#%-# %. '#1%58-- 3-&*,'# 9-,1#9 3-#*#' 2#5-A#%-H (1* 8*%. 3*2$--. #&25''( %'5+ *21 3'-. 3*2$--, -&8'5#%%(1 3 ,'. - %. 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'?5-$( 3-#*#% %-:# - 3-9#1 &*#52. 167 ) 6.1 | : X x1 x2 Y y1 y2 yj yl mx m11 m12 m1j m1l mx1 m21 m22 m2j m2l mx2 xi mi1 mi2 xk mk1 mk2 my my1 my2 mij mkj myj mil mxi mkl mxk myl n 2,' '#?2#',+ % ,%%-- (?- ?T#9 n "$#%-'" &%A#%-# 9*#5%. 42%$-- #8#,,-- - ,*#5' 3#*,&%-# 2,5%(1 9'#9'-A#,-1 :-*%-. ,52A.%. #5-A-%( Y, ,'#','2HO-1 3#*#5#%%(9 &%A#%-+9 ,52A.%. #5-A-%( X = x. 5+ 6'8 -,35&2H',+ 2%#%-+ #8#,,-- '8 * (693--A#,-# 2%#%-+ #8#,,--). -?5-:#%%# (:#%-# ($#%2) 9*#5%. 42%$-- #8#,,-- %&(H' & # ! yx = f (x a b : : : d). 6.4. 4? ! ! & '#-- #+'%,'#. -&#,'%, A' ,%%(9- 1'#-,'-9-, 3-,(HO-9- ,'#3#% ,+&- 9#:*2 ,,'5+HO-9- X - Y *29#%. ,52A.%. #5-A-%( (X Y), +5+H',+ # XY (#5+$-%%(. 99#%') 11 = XY = MZ(X ; mX )(Y ; mY )] = Z1 Z1 = (x ; mX )(y ; mY )f (x y) dx dy (6:26) ;1 ;1 - &# # = XY = MZ(X ; m X )(Y ; mY )] : (6:27) X Y X Y & (:#%-. (6.26) - (6.27) ,5#*2#', A' *5+ %1:*#%-+ XY - %#?1*-9 &%' &% ,3#*#5#%-+ *29#%. ,52A.%. #5-A-%(. ?5<-%,'# ,52A# 3- ??'# 6,3#-9#%'5%(1 *%%(1 &% ,3#*#5#%-+ *29#%. ,52A.%. #5-A-%( %#-&#,'#%. 6'92 *5+ $#%- '#,%'( ,+&- (-,. 6.5, 6.6 - 6.7) 3-9#%+H',+ 'A#A%(# $#%- ^XY - ^ #5-A-% XY - : & # n X 1 (6:28) ^XY = KXY = n (xi ; x)(yi ; y)U i=1 168 6Y s = ;0:9 =0 s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s = 0:9 s s s s - X 6.5 | ) + ( = ;0 9G 0 0G 0 9G + 1) & &# # ^ = r = KXY = (6:29) X Y X ! X !;1=2 n n n X 1 1 1 2 2 = n (xi ; x)(yi ; y) n (xi ; x) n (yi ; y) : i=1 i=1 i=1 *5%#.<#9 ?2*#' 35#&%. %#,5 -%+ 49 &3-,- 693--A#,8 #5+$-%%8 99#%': KXY = xy ; x y: (6:30) %58-A% 9:% &3-,' *5+ (?A%(1 *-,3#,-. s2x - s2y , A' n X s2x = n1 (xi ; x)2 = x2 ; x2 (6:31a) i=1 n X (6:31b) s2y = n1 (yi ; y)2 = y2 ; y2: i=1 2A#'9 6'-1 (:#%-. 352A#9 #O# *%2 49252 *5+ (A-,5#%-+ 693--A#,8 644-$-#%' #5+$--: h i h 2 2 i !;1=2 K 2 XY 2 : (6:32) ^ = r = = (xy ; x y) x ; x y ; y X Y -,'#9 %95%(1 2%#%-. 6'-1 ?&%A#%-+1 3-%-9#' -* (#8#,,-+ Y % X) 8 b0 + b1x = y < (6:33) : b x2 + b x2 = xy: 1 0 169 6Y s = ;0:9 =0 s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s = 0:9 s s s s - X 6.6 | ) + ( = ;0 9G 0 0G 0 9G +) #<+ 6'2 ,-,'#92, %1*-9 (6:34) b1 = xy2; x 2y = r ssY : x ;x X & ,-,'#9( (6.33) ,5#*2#', A' 3+9+ #8#,,-- Y % X 31*-' A##& 'A2 C = (x y). 5#*'#5%, 2%#%-# 3+9. #8#,,-- Y % X 9:% &3-,' -*# yx ; y = r ssY (x ; x): (6:35) X ,'23+ %58-A%(9 ?&9, 9:% 3&', A' ,-,'#9 %95%(1 2%#%-. #8#,,-- X % Y -9##' -* 8 a0 + a1y = x < (6:36) : a y2 + a y2 = xy: 1 0 #<+ 6'2 ,-,'#92, %1*-9 a0 = x ; b1y a1 = xy2; y 2x = r ssX : (6:37) y ;y Y 5#*'#5%, 2%#%-# 3+9. #8#,,-- X % Y 9:% &3-,' -*# xy ; x = r ssX (y ; y): (6:38) Y .*#%%(# 4925( 2*?%( , (A-,5-'#5%. 'A- &#%-+. #.,'-'#5%, A'?( &3-,' 2%#%-# #8#,,-- Y % X -5- X % Y, *,''A% %.'- 'A#A%(# $#%- %95%. *29#%. ,52A.%. #5-A-%( (X Y) : x, y, sx, sy , r. & 4925 (6.35) - (6.38) ('##' :%# ,'%<#%-#, ,+&(HO## 644-$-#%'( #8#,,-- a1, b1 - 644-$-#%' #5+$-- : q r2 = a1b1 -5r = a1b1: b0 = y ; b1x 170 y r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r ! r r r r r r r r r r r r r r r r r r r r r r r r r x 6.7 | C % : ) ( = 1)G ) ( 0 8)G ) ( 0 4)G ) ( 0 0) % 644-$-#%' #5+$-- ,3*#' , &%9 644-$-#%' #8#,,--. ,5- 49251 (6.35) - (6.38) 644-$-#%' #8#,,-- Y % X #% b1 = rsY =sX -, ,'#','#%%, 644-$-#%' #8#,,-- X % Y #% a1 = rsX =sY 1 (# ), ' 8+', A' %35#%-# &-,-9,'- Y % X 1 (# ). D' &%A-', A' 3##9#%%(# Y - X *%#9#%% $ ($). 644-$-#%'( a1 - b1 %# 3&5+H' ,2*-' ,'#3#%- ,+&- 9#:*2 ,52A.%(9#5-A-%9- Y - X. '#3#% ,+&- &-,-' ' 285, ?&%%8 3+9(9- #8#,,--. R#9 9#%<# 285 9#:*2 3+9(9- #8#,,--, '#9 '#,%## ,+& 9#:*2 ,52A.%(9#5-A-%9- Y - X. - ,5-+%-- *21 3+9(1 #8#,,-- *%2 -9##' 9#,' 5-%#.%+ 42%$-%5%+ &-,-9,' 9#:*2 Y - X. 85 9#:*2 5-%-+9- #8#,,-- 3#*#5+#',+ 2%#%-+9- (6.35) - (6.38). tg = b1 = rsY =sX - tg = a1 = rsX =sY , ' -& -,. 6.8 ('##', A' tg = tg 2 ; ; = ctg ( + ), - 36'92 ; tg tg = 1 ; r2 sX sY : (6:39) tg = 1 tg + tg r s2X + s2Y (A-,5-'#5%. 3'-# 3-%+' 35&',+ 644-$-#%'9 #5+$-p ^ = r = a1b1 = KXY =(sX sY ) A#,'# 3&'#5+ ,'#3#%- ,+&-. &9:%(# &%A#%-+ 644-$-#%' #5+$-- ^ 8%-A#%( -%'#59 Z;1U 1]. %2'- 6'8 -%'#5 #5-A-% ^ 3#*#5+#',+ -& ,5#*2HO-1 2,5-. : 1) #,5- 9#:*2 3##9#%%(9- X - Y ,2O#,'2#' 5-%#.%+ 35:-'#5%+ ,+&, ' 644-$-#%' #5+$-- ^ = 1U 2) #,5- 9#:*2 3##9#%%(9- X - Y ,2O#,'2#' 5-%#.%+ '-$'#5%+ ,+&, ' 644-$-#%' #5+$-- ^ = ;1U 3) 3- ',2','-- &-,-9,'- 9#:*2 3##9#%%(9- X - Y 644-$-#%' #5+$-- ^ = 0U 4) ,#1 ,'5%(1 ,52A+1 ;1 < ^ < 1. 3#*5- - ?'%# 2'#:*#%-#: A#9 ?5-:# 3 9*25H 644-$-#%' #5+$-- ^ #*-%-$#, '#9 ,-5%## 9#:*2 ,52A.%(9#5-A-%9- X - Y, A#9 ?5-:# %25H, '#9 6' &-,-9,' ,5?##. 171 y y 0 xy = a + a y yx = b + b x ~ 0 1 0 1 I C (x y) x 0 x 6.8 | : 1 + % + 39%-9, A' 644-$-#%' #5+$-- ^ 1'#-&2#' ,'#3#% 5-%#.%. &-,-9,'-, 36'92 *:# 3- ^ = 0 9:#' &',+, A' 9#:*2 X - Y ,2O#,'2#' 42%$-%5%+ ,+& %#5-%#.%8 -*. 85. 644-$-#%' 3+9. #8#,,-- Y % X %&(H' &# ! Y % X - ?&%AH' A##& yjx . %58-A% 285. 644-$-#%' 3+9. #8#,,-- X % Y %&(H' &# ! X % Y - ?&%AH' A##& xjy . D'- 644-$-#%'( 982' ?(' (A-,5#%( 3 49259 " X ! X n n !1 X n !# 1 1 1 yjx = 2 n xiyi ; n xi n yi = xy ; 2x y (6:40a) i=1 i=1 i=1 X X " X ! X n n !1 X n !# 1 1 1 (6:40b) xjy = 2 n xiyi ; n xi n yi = xy ; 2x y : i=1 i=1 i=1 Y Y 6'-1 ?&%A#%-+1 2%#%-+ 3+9(1 #8#,,-. 3-%-9H' -* y ; y = yjx(x ; x) x ; x = xjy (y ; y): (6:41) 5&# # 3-&% X - Y %&(H' #5-A-%2 (6:42) r = r(X Y) = p xjy yjx = " X X n n !1 X n !# 1 1 1 = n xiyi ; n xi n yi : X Y i=1 i=1 i=1 *' 644-$-#%' 5-%#.%. #5+$-- *#' &# # kdet, '(. -&9#+#' *5H -$-- 93%#%'( X, ?T+,%+#92H 5-+%-#9 Y, - %?'. kdet = r2 = xjy yjx : (6:43) 172 ,5- 5-%-- #8#,,-- '5-A%( ' 3+9(1, ' 644-$-#%' 5-%#.%. #5+$-- %# *#' 35%8 3#*,'5#%-+ ,-5# ,+&- 9#:*2 ,'#','2HO-9- 3-&%9- X - Y. 6'9 ,52A# & 9#2 &-,-9,'- 3-&% Y ' 3-&% X 3-%-9H' # 3 yjx, '# +5+#',+ '%<#%-#9 ,#*%#8 *'-A#,8 '5%#%-+ 2,5%(1 ,#*%-1 yx ,#*%#92 *'-A#,92 '5%#%-H 3-&% Y k ! P m (Y ; Y )2 1=2 xj xj Yx (6:44a) yjx = = j=1 ! : Pn m (y ; Y )2 1=2 Y j =1 yj j %58-A% *-',+ # 3 xjy 3-&% X ' 3-&% Y n ! P m (X ; X )2 1=2 yj yj Xy j =1 xjy = = (6:44b) ! : Pk m (x ; X )2 1=2 X j =1 xj j &9:%(# &%A#%-+ #5+$-%%(1 '%<#%-. yjx - xjy 8%-A#%( -%'#59 Z0U 1]. %2'- 6'8 -%'#5 6'- #5-A-%( 3#*#5+H',+ -& ,5#*2HO-1 2,5-. : 1) #,5- yjx = 0 (-5- xjy = 0), ' 3-&%- Y - X %# #5-2H'U 2) #,5- yjx = 1, ' 3-&% Y ,+&% , 3-&%9 X 42%$-%5%. ,+&H, y = f (x)U 3) (35%+H',+ %##%,' yjx jr(X Y)j - xjy jr(X Y)j. 6.5. & ? ! ! ,%%+ $#5 #5+$-%%8 %5-& ,,'-' (+5#%-- ,+&- 9#:*2 ,52A.%(9- #5-A-%9- X - Y -, #,5- :#',+, A' 6' ,+& ,2O#,'2#', | 3#*#5#%-# ,'#3#%- ?5-&,'- 6'. ,+&- 42%$-%5%.. 3'-# 3- ??'# 6,3#-9#%'5%(1 *%%(1 644-$-#%' #5+$-- 8#%#5%. ,23%,'- %#-&#,'#%. #&25''9 6,3#-9#%' % ,%%-- (?%1*-',+ #8 'A#A%+ $#% (3-?5-:#%%# &%A#%-#) | &# # r = ^. ,5- (?A%(. 644-$-#%' #5+$-- r &5,+ %(9 %25H, 6' #O# %# &%A-', A' ,'#','2HO-. 644-$-#%' #5+$-- #% %25H, ,52A.%(# #5-A-%( X - Y %#&-,-9( ( ,52A# -1 %95%,'-), - %?', #,5- r 6= 0, ' 6' #O# %# &%A-', A' 6= 0, ,52A.%(# #5-A-%( 982' &',+ %#&-,-9(9-. 5+ '8 A'?( % ,%# ,''-,'-A#,8 %5-& '#'-' % 3,, %1*+',+ 5- ,52A.%(# #5-A-%( #5+$-%%. &-,-9,'-, %#?1*-9 173 3.0 2.5 f (^r ) 2.0 1.5 1.0 0.5 0.0 ..... ..... ... ... ... ... ... ... ... .. .. ..... .. ..... . . . . . .. ... ... ... ... .. ... ... .. ... ... .... ... ..... ... . ..... . ... . ... ... ... ... ... ... ... ... ... ... ... ... .... . ... . . . ... . . ... . . . ... . . ... . . . ... . . ... . . . . ... . . ... . . . . ... ... . .. . . . ... ... . . . .... .. ... .. ... . .... ... . . . . ... . ... . . . . ... . . ... . ... . ... . . . ... . ... . .... . .. . ... . ... ... . . ... . . . . ... ... . ... . . . ... ... . . .... . ... ... . . .. . . ... ... . . . ... . . ... . .. . ... . . ... . . ... .... . ... . . ... . .... ... . ... . .. ... . . ... . . . . . . . . . ... ........................ . . . .... . . ... ..... ....... ... . ...... . . ....... ... . ....... ... .. . ..... . . . . . . ..... ... . .... . . . . . . . . ..... . . .... ... . ... . ... . . . . . . . . . ... . . ... .... ... . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . .... ........ ........ ... ... . ... . . ... . . . ... . . . . . . . . . ....... ... ......... ... . . . . ... . .... . . . . . . ... ..... . .... . . . . . . . . . ... . . . . . . . . . . . ... .... .... . . . ... . . . . ... . . . . . . . . . . . . . . .... .... ... . ... . . .. .. ... . . . . . . . . . . . . . . ..... ... .... ... . . . ... . .. . . . . . . .... . . . . . ... . .... ..... ... . . ... . . . . . . .. . . . . . . . . . ... ... ..... .... ..... .. ... ... .. .. . . . . . . . . ... .... ... ... ...... .... .. . . . .... . . . . . . . ... ..... ... . . ..... ...... . . . . . . . ... ... . . . . ..... . ... . ...... .. ... .. . . . . . . ... . . ... . . . . . .... .... ...... ... . . . . .. . . . . ... . . . . . . . . . .... ....... . ..... ... .. . . .. . . .. .... . . . . . . . . . . . . ..... ........ ... .... ..... ... .. . .. .. . . . . . . . . . . . . . . . . . . . . . . . . ... .... ............. .............. ...... . . ... . . . . . . . . . . . . . . . . . . . . ........ .... .... ... ..... ................. ..... ................... .... ... .... ......... .............. . ............. . ....................................................................................................................................................................................................................................................................................................................................... = ;0:8 = 0:8 = ;0:4 = 0:4 = 0:0 ;1:0 ;0:5 0.0 0.5 r^ 1.0 3.0 2.5 2.0 1.5 f (^r) 1.0 0.5 0.0 6.9 | ' 5 f (^r) = f (^rG n)G (H n = 10G / R## , / = {0,8G {0,4G 0,0G 0,4G 0,8) 3#-' %25#2H 8-3'#&2 fH0 : = 0g 3'- *%. -& 5'#%'-%(1 8-3'#& fH : 6= 0g, -5- fH : > 0g, -5- fH : < 0g. ,%, A' *5+ 3#- %25#. 8-3'#&( %#?1*-9 &%' &% ,3#*#5#%-+ (?A%8 644-$-#%' #5+$--. 5'%,' ,3#*#5#%-+ #+'%,'#. 693--A#,8 ((?A%8) 644-$-#%' #5+$-- r ,52A#, #,5- (? 352A#% -& ,23%,'- , *29#%(9 %95%(9 ,3#*#5#%-#9, &-,-' ' ?T#9 (?- n - 644-$-#%' #5+$-- 8#%#5%. ,23%,'- - -9##' -* : Z1 xn;2 n ; 2 dx : 2 ( n ; 1) = 2 2 ( n ; 4) = 2 p f (^r) = (1 ; ) (1 ; r^ ) (6:45) n;1 1 ; x2 (1 ; r ^ x ) 0 5+ (? -& 8#%#5%. ,23%,'-, '. #5-A-%( X - Y %95%( - %#&-,-9(, '.#. = 0, 35'%,' ,3#*#5#%-+ #+'%,'#. 693--A#,8 644-$-#%' #5+$-- 3#*#5+#',+ (:#%-#9 n;1 ; f (^r) = 1 n;2 2 (1 ; r^2)(n;4)=2: (6:46) ; 2 J925( (6.45) - (6.46) %&(H' ,3#*#5#%-#9 9#. -,. 6.9 - 6.10 3-#*#%( 84-- *-44##%$-5%. - -%'#85%. 42%$-. ,3#*#5#%-+ (?A%8 644-$-#%' #5+$-- 3- n = 10 = ;0:8, ;0 4U 0 0U 0 4U 0 8. 174 1.00 0.75 F (^r ) 0.50 0.25 = ;0:8 .......................................................................................................................................................................................... . ............................................. . ........... ....... .............. ...... ... .............. .......... ........... . . ..... . .......... ...... ... . . . . ... ... ........ ...... .. . . . . . . . . . . . . . . . . . . . .. .. .. .. ..... .... ...... ... ... .... ...... .... ... .... .... ... ... .... .... ... .. ... . . . . . . . . . . . . .. .... ... ... ... .. ... .... .. .. ... ... .... .. ... .... .... ... ... .. . . . . . . . . . . . . . . .. .. ... ... ... ... ... ... ... ... .. ... ... ... ... ... ... ... . . . . . . . . . . . . .. ... ... ... ... ... ... ... ... ... .. ... .. ... ... ... .. .. . . . . . . . . . . . .. .. ... ... ... .. ... .. ... ... .. ... .. ... ... ... ... ... . . . . . . . . . . . .. .. .. ... ... .. ... ... .. ... ... .. .. ... ... ... ... .. ... . . . . . . . . . . . .. .. .. ... ... .. ... ... ... ... .. ... ... ... ... .. ... .. . . . . . . . . . . . .. ... ... ... ... .. ... ... ... ... ... ... .. ... ... .. ... .. ... . . . . . . . . . .. .. .. ... ... .. .. .. ... ... ... ... .. ... ... ... .. ... .. . . . . . . . . . . . . . . .. ... ... ... ... ... ... ... ... ... .. ... ... ... ... ... ... ... . . . . . . . . . . . .. . .. ... .... .. .. ... .... .... ... ... .... ... .... .... ... ... ... . . ... . . . . . . . . . . . . .. .. ... .... ...... .. ... .... ..... ..... ... ... .... ..... ....... .... ... ...... ...... ........ . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .... ........ ........... .......... ... ...... ............ ................ ................. .................................................................................................................................................................................................................................... 0.00 ;1:0 = ;0:4 = 0:0 = 0:4 = 0:8 ;0:5 0.0 r^ 0.5 1.00 0.75 0.50 F (^r ) 0.25 1.0 0.00 6.10 | ' 5 +/ F (^r) = F (^rG n) (H n = 10G / R## , / = {0,8G {0,4G 0,0G 0,4G 0,8) ,'#%-# -%'#5%(1 $#% *5+ 644-$-#%' #5+$-- 8#%#5%. ,23%,'- 9#'*9-, %#3,#*,'#%% -,35&2HO-9- 4925( (6.45){(6.46), 3-9#%+#',+ *,''A% #*. 3'-# *5+ 3,'#%-+ -%'#5%(1 $#% *5+ 35&2H',+ 5-? %98999- ( ,3#$-5-&-%%(1 9'#9'-A#,-1 -5- ,''-,'-A#,-1 3#'1), ,%%%(9- % 6'-1 49251, 5-? 3-9#%+H' ,3#$-5%(# 3#?&%-+ 693--A#,8 644-$-#%' #5+$--, '(# 3&5+H' ,#,'- ,3#*#5#%-# %#'. 42%$-- ' (?A%8 644-$-#%' #5+$-- 1< -&2A#%%(9 ,3#*#5#%-+9, %3-9#, %95%92 -5,3#*#5#%-H 'H*#%'. # &%A-9,'- (?A%8 644-$-#%' #5+$-- r, '.#. 3#, 2H #5-A-%2 (?A%8 644-$-#%' #5+$-- ,5#*2#' ,A-'' *,''A%. *5+ ,''-,'-A#,- ?,%%%8 (* %5-A-- #5+$-%%. ,+&- 9#:*2 -,,5#*2#9(9- 3##9#%%(9- X - Y, ,%(#',+ % ,5#*2HO-1 '#1 9'#9'-A#,-1 9*#5+1, '(# +5+H',+ ??O#%-#9 -,1*%(1 3#*3,(5 *29#%. 8#%#5%. ,23%,'- (X Y). 4 1. -9#%+#',+ *5+ 3#- 8-3'#&( ? ',2','-- #5+$-%%. ,+&9#:*2 -,,5#*2#9(9- 3##9#%%(9-, '.#. *5+ 3#- 8-3'#&( fH0 : = 0g. ,1*%(# 3#*35:#%-+ - 8%-A#%-+ : ) *29#%(. &% ,3#*#5#%-+ -,,5#*2#9(1 3##9#%%(1 (X Y) 8#%#5%. ,23%,'- 3#*358#',+ %95%(9U 175 ?) ?T#9 (?- n 9:#' ?(' 5H?(9. 5+ 3#- &%A-9,'- (?A%8 644-$-#%' #5+$-- (A-,5+#',+ ,''-,'- q (6:47) t = r (n ; 2)(1 ; r2) -9#HO+ ,3#*#5#%-# 'H*#%' , = n ; 2 ,'#3#%#. ,?*(. 5+ 3#- %25#. 8-3'#&( fH0 : = 0g %1*+' 3 '?5-$9 ,3#*#5#%-+ 'H*#%' 3 4-,-%%92 2%H &%A-9,'- - A-,52 ,'#3#%#. ,?*( = n;2 -'-A#,# &%A#%-# t=2 n;2, 2*5#'+HO## 2,5-H Pr(jtj t=2 n;2 ) = . ,5jtj t=2 n;2 (6:48a) ' %25#2H 8-3'#&2 ? ',2','-- 5-%#.%. &-,-9,'- 9#:*2 3##9#%%(9- X Y ,5#*2#' '#8%2'. ,5- :# jtj < t=2 n;2 (6:48b) ' %#' ,%%-. '#8' %25#2H 8-3'#&2 %##5-%%,'- 3##9#%%(1 X - Y. 4 2. -9#%+#',+ *5+ 3#- 8-3'#&( ,-5# #5+$-%%. ,+&- 9#:*2 3##9#%%(9- X - Y, -%A# 8+, *5+ 3#- %25#. 8-3'#&( '9, A' 644-$-#%' #5+$-- 8#%#5%. ,23%,'- #% %#'92 4-,-%%92 A-,52, '.#. fH0 : = 0g. (,5- 0 = 0, ' 3#+#',+ 8-3'#& %##5-%%,'- 3##9#%%(1 X - Y.) ,1*%(# 3#*35:#%-+ - 8%-A#%-+ : ) *29#%(. &% ,3#*#5#%-+ -,,5#*2#9(1 3##9#%%(1 (X Y) 8#%#5%. ,23%,'- 3#*358#',+ %95%(9U ?) ?T#9 (?- n *,''A% #5- (n 50)U ) (A-,5#%%# &%A#%-# (?A%8 644-$-#%' #5+$-- %##5- (jj 0 5). ,5- 6'- 3#*35:#%-+ (35%#%(, ' (?A%(. 644-$-#%' #5+$-r -9##' 3-?5-:#%% %95%# ,3#*#5#%-# , 9'#9'-A#,-9 :-*%-#9, %(9 644-$-#%'2 #5+$-- 8#%#5%. ,23%,'- - *-,3#,-#., %. p 2 r = (1 ; r ) n. D' %#3,#*,'#%% 9:% &9#'-', %5-&-2+ 84- 35'%,',3#*#5#%-+ (?A%8 644-$-#%' #5+$-- (-,. 6.6). ',H* ,5#*2#', A' #,5- %25#+ 8-3'#& fH0 : = 0g #%, ' ,''-,'- (6:49) u = q r ; 20 p (1 ; 0)= n -9##' 3-?5-:#%% %95%# ,3#*#5#%-# , %25#(9 9'#9'-A#,-9 :-*%-#9 - *-,3#,-#., %. #*-%-$#. (?##9 u=2 | -'-A#,# &%A#%-# ,'%*'-&%%. %95%. ,52A.%. #5-A-%(, 2*5#'+HO#. 2,5-H Pr(juj u=2) = . 176 ,5- juj u=2 (6:50a) ' 8-3'#& fH0 : = 0g '#8#',+. ,5- :# juj < u=2 (6:50b) ' %#' ,%%-+ '#8' %25#2H 8-3'#&2. -9#%#%-# 6'. 9*#5- 3&5+#' ':# %1*-' 3-?5-:#%%(# *#-'#5%(# -%'#5( *5+ 644-$-#%' #5+$-- 8#%#5%. ,23%,'- 3 4925# q q p p 2 2 r ; u=2 (1 ; 0)= n < < r + u=2 (1 ; 0)= n : (6:51) ,5- :#',+, A' (A-,5#%%(. *#-'#5%(. -%'#5 %# %(#' &%A#%-# = 0, ' 8-3'#& fH0 : = 0g '#8#',+, 3'-%9 ,52A# *%%(# 6,3#-9#%' %# 3&5+H' '#8%2' %25#2H 8-3'#&2. 4 3. ,1*%(# 3#*35:#%-+ - 8%-A#%-+ : ) *29#%(. &% ,3#*#5#%-+ -,,5#*2#9(1 3##9#%%(1 (X Y) 8#%#5%. ,23%,'- 3#*358#',+ %95%(9U ?) ?T#9 (?- n *,''A% #5- (n 50). 5+ 3#- 8-3'#&( ,-5# #5+$-%%. ,+&- (A-,5+#',+ ,''-,'- (3#?&%-# J-<#) 1 + r 2z ; 1 1 (6:52) z = 2 ln 1 ; r ':#: r = thz = ee2z + 1: :% 3&', A' ,3#*#5#%-# ,''-,'-- z *,''A% 1< 33,-9-2#',+ %95%(9 ,3#*#5#%-#9 , 9'#9'-A#,-9 :-*%-#9 ! 1 1 + 0 MZZ ] = 2 ln 1 ; + 2(n;0 1) (6:53) 0 - %# &-,+O#. ' *-,3#,-#. DZZ ] = n ;1 3 : (6:54) ''-,'- !p 1 + r 1 + 0 0 u = 1 1513 lg 1 ; r ; 1 1513 lg 1 ; ; 2(n ; 1) n ; 3 (6:55) 0 -9##' ,-93''-A#,- %95%# ,3#*#5#%-# , %25#(9 9'#9'-A#,-9 :-*%-#9 - *-,3#,-#., %. #*-%-$#. # 8-3'#&( fH0 : = 0g 3*-',+ 3 '. :# ,1#9#, A' - *#5- 2, '.#. 3 4925# (6.55) (A-,5+#',+ 693--A#,# &%A#%-# u ,''-,'-- u. 5&2+, '?5-$#. 42%$-- K35,, 3 4-,-%%92 2%H &%A-9,'- %1*+' -'-A#,# &%A#%-# u=2, 2*5#'+HO## 2,5-H Pr(juj u=2) = . 177 ,5- (A-,5#%%# &%A#%-# ,''-,'-- 2*5#'+#' %##%,'2 juj u=2 (6:56a) ' 8-3'#&2 fH0 : = 0g ,5#*2#' '#8%2'. ,5- :# juj < u=2 (6:56b) ' %#' ,%%-+ '#8' %25#2H 8-3'#&2. 2 . *#5 3 +5+#',+ %-?5## ?O#., ' 9:#' 3-9#%+',+ 35H?(1 &%A#%-+1 n, - r. 8* 3#+#',+ %25#+ 8-3'#& fH0 : = 0g, ' $#5#,?&%## -,35&' *#5 1. ,5- :# (A-,5#%%# &%A#%-# (?A%8 644-$-#%' #5+$-- %# +5+#',+ A#% ?5<-9 (jrj < 0 5) - ?T#9 (?*,''A% #5- (n 50), ' ,5#*2#' (?' *#5 2. 6.6. :! $# $ ! ? ! & & ,5# %1:*#%-+ 693--A#,8 2%#%-+ #8#,,-- Y % X yx = b0 + b1x 3*,A-'(H' ,#*%HH *'-A%2H <-?2 se sy:x, 1'#-&2HO2H ,'#3#% ,,#-%-+ 6,3#-9#%'5%(1 'A# 28 5-%-- #8#,,--, '.#. ,'#3#% ,,#-%-+ &-,-9. 3##9#%%. Y, A-O#%%. ' 5-+%-+ 3##9#%%. X : !1=2 n X 1 2 se sy:x = n ; 2 (yi ; yx) : (6:57) i=1 #*%-# *'-A%(# <-?- b0 - b1 3#*#5#%-+ 644-$-#%' b0 - b1 3#*#5+H',+ (:#%-+90 "X #;1 1 n 1 2 @ + x2 2 A ( x (6:58a) b20 = y:x i ; x) n b21 = 2 y:x "X n i=1 i=1 (xi ; x)2 #;1 : (6:58b) 2 #E %#,9#O#%%. $#%. 9#%++ 6'-1 49251 %#-&#,'%2H *-,3#,-H y:x 3,5# 3#?&%-. 352A-9 693--A#,-# *-,3#,-- 644-$-#%' 0 - 1 5-%#.%8 2%#%-+ #8#,,-- se sy:x 2 n 3 X X n ! X n !2 ;1 b20 = s2y:x x2i 4n x2i ; (6:59) xi 5 s2y:x, i=1 i=1 178 i=1 2 n 3 X n !2 ;1 X b21 = ns2y:x 4n x2i ; xi 5 : i=1 (6:60) i=1 *'%(. #% -& 6'-1 &%A#%-. %&(#',+ 3 %1 # b0 b1. #*%++ *'-A%+ <-? 2&(#', %,5 ,#*%#9 $#%- 644-$-#%' b0 - b1 '5-AH',+ ' 9*#5%(1 644-$-#%' #8#,,-- 0 - 1. #'*( #8#,,-%%8 %5-& &(H',+ ,*#:'#5%(9- - #'%(95-< 3- ,?5H*#%-- %#'(1 '#?%-., 3#*T+5+#9(1 2,5-+9 ,? ,''-,'-A#,8 9'#-5 6,3#-9#%'#. ,5- 8-3'#& %95%9 &%# ,3#*#5#%-+ ,-,'#9( ,52A.%(1 #5-A-% (X Y) ,3#*5-, ' : ) 9'#9'-A#,# :-*%-# MZYjX = x] = 0 + 1xU ?) 2,5%+ *-,3#,-+ DZYjX = x] 3,'+%% *5+ ,#1 &%A#%-. x - % 2 y:x = y2(1 ; 2)U ) ,3#*#5#%-# fY (yjx) %95%#U 8) %?5H*#%-+ f(xi yi)g, (i = 1 2 : : : n) %#&-,-9(. ,5- &(#',+ &9:%(9 3-%+', A' %95%+ 9*#5 +* %?5H*#%-. f(xi yi)g 1< ':#' &%9#%,'- -,,5#*2#9. *29#%. #5-A-%( (X Y), '.#. 3##A-,5#%%(# '#?%-+ ) | 8) (35%+H',+, ' 6'9 ,52A# $#%644-$-#%' #8#,,-- b0 - b1 3'-A#,-1 ,A#'1 3-,(H' (3-?5-:#%%) %95% ,3#*#5#%%(9- ,52A.%(9- #5-A-%9- , 9'#9'-A#,-9- :-2 -&#,'%, ' ,''-,'-*%-+9- 0, 1 - *-,3#,-+9- b20 , b21 . 8* #,5- y:x 2 2 u0 = (b0 ; 0)= y:x - u1 = (b1 ; 1)= y:x ,3#*#5#%( 3 %95%92 &%2 , %25#(9 9'#9'-A#,-9 :-*%-#9 - *-,3#,-#., %. #*-%-$#. ''-,'-- uj , j = 0 1, 9:% 3-9#%+' *5+ 3,'#%-+ -%'#5%(1 $#% 644-$-#%'. 6'9 ,52A# -,9(# *#-'#5%(# -%'#5( *5+ 644-$-#%' -,'-%%8 2%#%-+ #8#,,-- ?2*2' %1*-',+ 3 &*%%. *#-'#5%. #+'%,'- p = 1 ; 3 4925# ! b j ; j j = 0 1 (6:61) Pr ;u=2 < pc < u=2 = 1 ; y:x jj 8*# u=2 | %'-5 %95%8 ,3#*#5#%-+ N (0U 1)U Pn x2 i n c00 = Pn i=1 Pn 2 c11 = Pn Pn 2 : 2 2 n xi ; xi n xi ; xi i=1 i=1 i=1 (6:62) i=1 #?&2+ (:#%-# (6.61), 352A-9 (j = 0, 1) Pr bj ; u=2 y:xpcjj < j < bj + u=2 y:xpcjj = 1 ; : (6:63) - 3#*#%-- 6,3#-9#%'5%(1 ?' #5-A-% y:x %#-&#,'%. #*35:-9, A' , 39OH 9#'* %-9#%<-1 *' ?(5 %.*#% %#,9#O#%%+ 179 $#% ^y:x #5-A-%( y:x 3 4925# ^y:x = sy:x n X 1 = n ; 2 (yi ; b0 ; b1x)2: i=1 (6:64) 8* ,''-,'- (j = 0 1) t = bj s; j = sbj ;pc j (6:65) bj y:x jj -9##' ,3#*#5#%-# 'H*#%' , = n ; 2 ,'#3#%+9- ,?*(, '2H 9:% -,35&' *5+ 3,'#%-+ -%'#5%(1 $#% 644-$-#%' 5-%#.%8 2%#%-+ #8#,,--. ,,9'-9 100(1 ; ) %{. *#-'#5%(. -%'#5 *5+ 644-$-#%' 0 1. 5+ #8 3,'#%-+ 3 '?5-$9 t-,3#*#5#%-+ 'H*#%' 3 A-,52 ,'#3#%#. ,?*( = n ; 2 - *#-'#5%. #+'%,'- p = 1 ; %1*-9 &%A#%-# t=2 n;2, 2*5#'+HO## 2,5-H Pr jtj < t=2 n;2 = 1 ; : (6:66) ,5# 3,'(1 3#?&%-. 352A-9 ! b 0 ; 0 (6:67) Pr ;t=2 n;2 < s pc < t=2 n;2 = 1 ; y:x 00 ! b 1 ; 1 Pr ;t=2 n;2 < s pc < t=2 n;2 = 1 ; : (6:68) y:x 11 #&25''# 3#?&%-+ ?-1 %##%,' 285(1 ,?1 %.*#9 *5+ 100(1 ; ) %{8 *#-'#5%8 -%'#5 *5+ 644-$-#%' 5-%#.%. #8#,,-- 0 - 1 : b0 ; t=2 n;2 sy:xpc00 < 0 < b0 + t=2 n;2sy:x pc00 (6:69) (6:70) b1 ; t=2 n;2 sy:xpc11 < 1 < b1 + t=2 n;2sy:x pc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mX = xf (x y) dx dy = = R xf (x) dx RR MZY] = mY = yf (x y) dx dy = R = yf (y) dy RR X2 = (x ; mX )2f (x y) dx dy = = R (x ; m )2f (x) dx (?A%+ ,23%,' x = 1 P n i xi y = 1 P n j yj s2X = X Y2 = R R (y ; m )2f (x y) dx dy = Y R = (y ; m )2f (y) dy Y XY = = R R xyf (x y) dx dy ; MZX]MZY] = XY X Y s2Y = 1 P 2 n i (xi ; x) = x2 ; x2 1 P 2 n j (yj ; y) = y 2 ; y2 = = KXY = = n1 P P (xi ; x) (yj ; y) i j r = KXY sX sY *#5%# 2%#%-# #8#,,-- Y % X MZYjX = x] = mY + XY (x ; mX ) D93--A#,# 2%#%-# #8#,,-- Y % X yx = y + r ssXY (x ; x) *#5%# 2%#%-# #8#,,-- X % Y MZXjY = y] = mX + XY (y ; mY ) D93--A#,# 2%#%-# #8#,,-- X % Y xy = x + r ssYX (y ; y) ' 1 '?5-$# 3-#*#%( #&25''( n = 11 -&9##%-. &%A#%-. xi %9-%5 *5-% 9*#5#. (3-&% X) - <--% 9*#5#. yi (3-&% Y) : 181 i 1 2 3 4 5 6 xi, 0,90 1,22 1,32 0,77 1,30 1,20 yi, {0,30 0,10 0,70 {0,28 {0,25 0,02 i 7 8 9 10 11 xi, 1,32 0,95 1,45 1,30 1,20 yi, 0,37 {0,70 0,55 0,35 0,32 5+ 6'-1 *%%(1 3#*358#',+, A' 9#:*2 3-&%9- X - Y ,2O#,'2#' 5-%#.%+ #8#,,-%%+ &-,-9,' MZYjX = x] = 0 + 1x. #?2#',+ : 1) $#%-' 39#'( 0 - 1 9*#5%8 2%#%-+ #8#,,-- 9#'*9 %-9#%<-1 *'U 2) %.'- ,#*%-# *'-A#,-# <-?- 644-$-#%' %.*#%%8 693--A#,8 2%#%-+ #8#,,--U 3) 3,'-' 95%{# *#-'#5%(# -%'#5( *5+ 644-$-#%' 5-%#.%8 2%#%-+ #8#,,-- 0 - 1U 4) 35&2+, 693--A#,-9 2%#%-#9 #8#,,--, %.'- 'A#A%2H $#%2 '5%#%-+ ' %9-%5%8 &9# *5-%( 9*#5-, #,5- <--% 9*#5- '5%+#',+ ' %9-%5%8 &9# % #5-A-%2 x = 1 1 U 5) (A-,5-' 644-$-#%' #5+$-- - 644-$-#%' *#'#9-%$--. ?T+,%-' ,9(,5 644-$-#%' *#'#9-%$--. () 1. %A5 (A-,5-9 ,398'#5%(# ,299(. ,5#*'#5% %1*-9 : n X xi = 12 93U n X i=1 i=1 xiyi = 1 7193U n X i=1 n X i=1 yi = 0 88U x2i = 15 6411U n X i=1 yi2 = 1 8856: .*#9 (?A%(# ,#*%-# : 1 12 93 = 1 175U 1 0 88 = 0 08: x = 11 y = 11 .*#9 $#%- b0 - b1 644-$-#%' -,'-%%8 2%#%-+ #8#,,-- 3 49259 U Pn x y ) ; (Pn x ) (Pn y ) 11 1 7193 ; 12 93 0 88 n ( i=1 i i i=1 i i=1 i = b1 = P P n n 2 11 15 6411 ; 12 932 = 1 5479U n ( i=1 xi ) ; ( i=1 xi)2 b0 = y ; b1x = 0 08 ; 1 5479 1 175 = ;1 7394: 5#*'#5%, 693--A#,# 2%#%-# #8#,,-- -9##' ,5#*2HO-. -* : yx = ;1 7394 + 1 5479x: 74- 3,'#%%8 693--A#,8 2%#%-+ 5-%#.%. #8#,,-- 3-#*#% % -,. 6.11. 182 1 0.8 y 0.6 0.4 0.2 0 -0.2 s -0.4 -0.6 -0.8 -1 0 0.2 0.4 0.6 0.8 s s s ss s s s s s x 1 1.2 1.4 1.6 1.8 2 6.11 | >+ R/ G R + y x = ;1 7394 + 1 5479x 2. 5+ %1:*#%-+ ,#*%-1 *'-A#,-1 <-? b - b , 1'#-&2H0 1 O-1 'A%,' %.*#%%(1 644-$-#%' b0 - b1 693--A#,8 2%#%-+ #8#,,--, (A-,5-9 %A5# ,#*%HH *'-A#,2H <-?2, 1'#-&2HO2H ,,#-%-# 693--A#,-1 'A# 28 5-%-- #8#,,--. 5+ 6'8 3-&#*#9 $#%2 &%A#%-. &-,-9. 3##9#%%. 3 4925# yx = ;1 7394 + 1 5479xi - (A-,5-9 '5%#%-+ ei = yx + 1 7394 ; 1 5479xi . #&25''# %.*#9 P11 i=1 ei = 0 7561. 3#*#5-9 %#,9#O#%%2H $#%2 *-,3#,-- &-,-9. 3##9#%%. Y, A-O#%%. ' 5-+%-+ 3##9#%%. X, 3 4925# n 1 X 0 7561 = 0 08401: 2 = s2 = 2 ^y:x ( y + 1 7394 ; 1 5479 x i) = y:x x n ; 2 i=1 9 (A-,5-9 693--A#,-# *-,3#,-- 'A#A%(1 $#% 644-$-#%' #8#,,-- : 2 n 3 X X n ! X n !2 ;1 s2b0 = s2y:x x2i 4n x2i ; xi 5 = i=1 i=1 i=1 6411 0 0841 = 0 2689U = 11 15 15 6411 ; 12 932 3 2 n X n !2 ;1 2 X 2 2 xi 5 = sb1 = nsy:x 4n xi ; i=1 i=1 0 0841 = 11 1511 6411 ; 12 932 = 0 1899: 5#*'#5%, sb0 = 0 520U sb1 = 0 436. 183 3. 5+ 3,'#%-+ 95%{1 *#-'#5%(1 -%'#5 *5+ 644-$-#%' 5-%#.%8 2%#%-+ #8#,,-- 0 - 1 3 '?5-$9 t-,3#*#5#%-+ 'H*#%' 3 A-,52 ,'#3#%#. ,?*( = n ; 2 = 9 - *#-'#5%. #+'%,'- p = 1 ; = 0 95 %1*-9 -'-A#,# &%A#%-# ,''-,'-- (%'-5) t=2 n;2 = t00259 = 2 262. ,35&2+ 49252 b0 ; t=2 n;2sb0 < 0 < b0 + t=2 n;2sb0 %1*-9 95%{. *#-'#5%(. -%'#5 *5+ 644-$-#%' b0 b0 t00259sb0 = ;1 7394 (2 262 0 521) = Z;2 9194U ;0 5594] : 5#*'#5%, ;2 9194 < b0 < ;0 5594. %58-A%, -,35&2+ 49252 b1 ; t=2 n;2sb1 < 1 < b1 + t=2 n;2sb1 %1*-9 95%{. *#-'#5%(. -%'#5 *5+ 644-$-#%' b1 b1 t00259sb1 = +1 5479 (2 262 0 436) = 1 5479 0 9885: 5#*'#5%, ;0 5594 < b1 < 2 5364. 4. .*#9 'A#A%2H $#%2 '5%#%-+ ' %9-%5%8 &9# *5-%( 9*#5-, #,5- <--% 9*#5- '5%+#',+ ' %9-%5%8 &9# % #5-A-%2 x = 1 1 : yx = 1 7394 + 1 5479 1 10 = ;0 036 : 5. (A-,5-9 644-$-#%' #5+$-Pn x y ) ; (Pn x ) (Pn y ) n ( r = q Pn 2 i=1 Pin i 2 qi=1 Pi n i2=1 i Pn 2 = n ( i=1 xi ) ; ( i=1 xi) n ( i=1 yi ) ; ( i=1 yi) 11 1 7193 ; 12 93 0 88 p = p = 0 7534: 2 11 15 6411 ; 12 93 11 1 8856 ; 0 882 5#*'#5%, 644-$-#%' *#'#9-%$-- ,5#*2HO-. : kdet = r2 = 0 584: 52A#%%(. #&25'' &%A#', A' 58,4 % ,,#-%-+ &-,-9. 3##9#%%. ?T+,%+#',+ 5-%#.%. #8#,,-#. Y % X, 41,6 % ,,#-%-+ Y 982' ?(' (&%( 5-? ,52A.%(9- <-?9- 6,3#-9#%', 5-? '#9, A' 5-%#.%+ #8#,,-%%+ 9*#5 351 ,85,2#',+ , 6,3#-9#%'5%(9- *%%(9-. ' 2 '?5-$# 3-#*#%( #&25''( n = 50 -&9##%-. &%A#%-. xi 3-&% X &%A#%-. yi 3-&% Y. 184 ) 6.2 | > 2 xi 81 77 76 86 53 47 36 40 49 60 yi xi yi xi yi 77 54 81 100 129 96 40 57 95 145 86 61 86 106 142 92 68 87 118 120 98 53 98 109 95 53 88 87 107 107 63 136 153 120 133 80 129 133 114 140 64 126 159 113 149 66 96 134 123 147 xi 94 84 73 107 94 107 99 100 104 88 yi xi yi 104 84 96 108 94 112 93 152 136 124 98 104 112 77 103 113 88 115 95 94 123 112 76 111 116 84 127 93 73 129 ,35&2+ &%A#%-+, 3-#*#%%(# *%%. '?5-$#, '#?2#',+ : 1) ,,'-' #5+$-%%2H '?5-$2U 2) %.'- 3 *%%(9 #5+$-%%. '?5-$( A-,5(# 1'#-,'-(?- x, y, sX , sY , KXY , rU 3) 3,'-' #5+$-%%# 35#U 3 1'#2 ,35:#%-+ 'A# % #5+$-%%9 35# 3*?' ?O-. -* 42%$-- #8#,,--U 4) %.'- 39#'( 693--A#,. 5-%#.%. 42%$-- #8#,,-- Y % X - X % Y - 3,'-' -1 84--. 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'9#A#%( ,##*-%( ,'#','2HO-1 -%'#5. ) 6.3 | : Y 40 160 140 120 100 80 1 60 4 nx 5 X 60 80 100 120 140 ny 1 3 4 3 5 8 4 8 1 13 2 7 4 13 3 3 7 1 5 6 14 15 7 3 n = 50 2. 5+ 3#*#5#%-+ 1'#-,'- -,9(1 693--A#,-1 2%#%-. #8#,185 ,-- %.*#9 : ,#*%-# -49#'-A#,-# n X x = n1 xi = 88 80U i=1 n X y = n1 yi = 109 60U i=1 (?A%(# *-,3#,-n X 1 2 sx = n (xi ; x)2 = 690 56U i=1 s2y n X 1 = n (yi ; y)2 = 771 84U i=1 (?A%(# ,#*%-# *'-A#,-# '5%#%-+ q q sx = 690 56 = 26 28U sy = 771 84 = 27 78U (?A%(. #5+$-%%(. 99#%' n n X X Kxy = n1 (xi ; x)(yi ; y) = xy ; x y = n1 xiyi ; x y = 651 52U i=1 i=1 (?A%(. 644-$-#%' #5+$-651 52 = 0 89: = r = sKxy 26 28 27 28 x sy #-9 &%A-9,' 352A#%%8 (?A%8 644-$-#%' #5+$--. 5+ 6'8 (A-,5-9 ,''-,'-2 : p p 50 ; 2 = 13 52: t = rp n ; 22 = 0p 89 1 ; 0 892 1;r .*#9 3 '?5-$9 %'-5#. ,3#*#5#%-+ 'H*#%' 3 A,' 23'#?5+#992 3'-# ,''-,'-A#,-1 ,A#' 2%H &%A-9,'- = 0 05 - A-,52 ,'#3#%#. ,?*( = n ; 2 = 48 %'-5 t=2 n;2 = t002548 = 2 02. ,52 &5,, A' t > t=2 n;2, ' 9:% ,*#5' (*, A' 5-%#.%+ #8#,,-%%+ 9*#5 MZYjX = x] = 0 + 0x (?% 2*A%, '.#. % ,85,2#',+ , 6,3#-9#%'5%(9- *%%(9-. 3. 5+ 3*'#:*#%-+ ,2O#,'%-+ 5-%#.%. #8#,,-%%. &-,-9,'9#:*2 -,,5#*2#9(9- 3##9#%%(9- X - Y 3,'-9 #5+$-%%# 35#. &?&-9 #&25''( -&9##%-. f(xi yi)g, (i = 1 2 : : : 50) -*# 'A# *#'. ,-,'#9# *-%' (-,. 6.12). -&25%+ $#% ,35:#%-+ 'A# % #5+$-%%9 35# 3&5+#' 3-%+' 8-3'#&2 5-%#.%. #8#,,-%%. &-,-9,'- 9#:*2 3-&%9- X - Y. 4. .*#9 &%A#%-+ 39#' 693--A#,8 2%#%-+ #8#,,-- 3-&% Y % 3-&% X 27 78 (x ; 88 80) = 25 68 + 0 94x yx = y + r ssy (x ; x) = 109 60 + 0 89 26 28 x 186 - &%A#%-+ 39#' 693--A#,8 2%#%-+ #8#,,-- 3-&% X % 3-&% Y 26 28 (y ; 109 60) = ;3 81 + 0 85y: xy = y + r ssx (y ; y) = 888 80 + 0 89 27 78 y %'5 (A-,5#%-. : a1b1 = 0 85 0 94 0 80 r2: 74-- %.*#%%(1 693--A#,-1 42%$-. 5-%#.%. #8#,,-- %%#,#%( % -,. 6.12. 200 y 180 160 s s 140 s 120 100 80 60 40 s s s 20 s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s ss s s s s x s 40 60 80 100 120 140 160 180 6.12 | : R/ G 0 + yx = 25 68+0 94xG + xy = ;3 81 + 0 85y 6.8. ' 6.1 9#H',+ *%%(# ,'-9,'- &'%-,58 %'-+ Na N O3 &-,-9,'' '#93#'2( *(. 100 A,'+1 *( ,'+#',+ ,5#*2HO## A-,5 2,5%(1 A,'#. Na N O3 3- ,'#','2HO-1 '#93#'21. tC 0 4 10 15 21 29 36 51 68 Na N O3 66,7 71,0 76,3 80,6 85,7 92,9 99,4 113,6 125,1 #*358+, A' 5-A#,' Na N O3 (,52A.%+ #5-A-% Y), '# ,'+#',+ 100 A,'+1 *(, &-,-' 5-%#.% ' '#93#'2( (,52A.%+ #5-A-% 187 X) ,', %.'- 39#'( a - b 4925# y = ax + b 3 9#'*2 %-9#%<-1 *'. () 5+ %1:*#%-+ 39#' a - b 3 9#'*2 %-9#%<-1 *' %#?1*-9 #<-' ,-,'#92 8 Pn Pn Pn > > < a i=1 x2i + b i=1 xi = i=1 xiyi n Pn > > = P yi : a xi + bn i=1 i=1 '. 6'9 ,52A# xi | '#93#'2 ,'U yi | 5-A#,' Na N O3, '# ,'+#',+ 100 A,'+1 *( 3- *%%. '#93#'2#. ,5# ,A#' 644-$-#%' ,-,'#9( 352A-9 8 10144a + 234b = 24628 6 < : 234a + 9b = 811 3: ',H* a = 0 87U b = 67 5. 5#*'#5%, &-,-9,' Y ' X -9##' -* y = 0 87 x + 67 5. ' 6.2 .'- 2%#%-+ #8#,,-- Y % X - X % Y 3 A#'(#9 39 %?5H*#%%(1 &%A#%-. ,52A.%. #5-A-%( (X Y) : xi 1 yi 2 2 4 3 5 4 7 () -,. 6.13 %%#,#%( 'A- f(xi yi)g *5+ i = 1 2 3 4. %5-&-2+ -1 ,35:#%-#, &9#A#9, A' %- 8233-2H',+ 28 3+9. 5-%--. 6'92 ?2*#9 3*?-' 2%#%-+ #8#,,-- 5-%#.%8 -*. #?1*-9(# (A-,5#%-+ ,35:-9 '?5-$#. i 1 2 3 4 299( xi yi xiyi x2i 1 2 2 1 2 4 8 4 3 5 15 9 4 7 28 16 10 18 53 30 yi2 4 16 25 49 94 *,'5++ ,299( ,-,'#92 %95%(1 2%#%-. (6.23), -9##9 8 4b0 + 10b1 = 18 < : 10b0 + 30b1 = 53: #<+ 6'2 ,-,'#92, %1*-9 : b0 = 0 5U b1 = 1 6. 188 8 6y y x = 0 5 + 1 6x t ....... .......... ......... ......... ......... .......... . . . . . . . . .. ......... ........ .......... ........ ......... .......... ........ . . . . . . . ......... ......... ........ .......... ........ .......... ......... . . . . . . . . ....... .......... ........ ......... ......... ......... ......... . . . . . . . . . . ........ ......... .......... ......... .......... ........ ......... . . . . . . . . ......... ........ 6 t t 4 2 t 0 1 2 3 4 x 6.13 | : 1 + Y X 5#*'#5%, 2%#%-# #8#,,-- Y % X -9##' -* yx = 0 5 + 1 6x: .*#9 '#3# 2%#%-# #8#,,-- X % Y -* (6.25). ,35&2+ &%A#%-+ 2:# %.*#%%(1 ,299, -9##9 8 4a0 + 18a1 = 10 < : 18a + 94a = 53: 0 1 #<+ ,-,'#92, %1*-9: a0 = ;7=26U a1 = 8=13. 5#*'#5%, 2%#%-# #8#,,-- X % Y -9##' -* xy = ;0 269 + 0 615y: ' 6.3 2,' '#?2#',+ %.'- 5-%#.%(# 2%#%-+ #8#,,-- Y % X - X % Y 3 10 39 %?5H*#%%(1 &%A#%-. ,52A.%. #5-A-%( (X Y) | 'A9 (xi yi) (i = 1 2 : : : 10). 9# -&9##%-+ i 1 2 3 4 5 6 7 8 9 10 xi 1 1 1 2 2 2 3 3 3 4 yi 3 3 3 4 4 5 5 5 6 7 () ,5- 3,'-' #5+$-%%# 35#, ' -&25% 9:% ,85,-',+ , '#9, A' 'A- (xi yi) (i = 1 2 : : : 10) 8233-2H',+ 5 3+9. 5-%--. 6'92 ?2*#9 3*?-' 2%#%-+ #8#,,-- 5-%#.%8 -*. (A-,5-9 ,299(, 1*+O-# ,-,'#9( %95%(1 2%#%-. (6.23) - (6.25), %# 3-&*+ 8233-- 6,3#-9#%'5%(1 *%%(1. 189 i 1 2 3 4 5 6 7 8 9 10 299( xi 1 1 1 2 2 2 3 3 3 4 22 yi 3 3 3 4 4 5 5 5 6 7 45 xiyi 3 3 3 8 8 10 15 15 18 28 111 x2i 1 1 1 4 4 4 9 9 9 16 58 yi2 9 9 9 16 16 25 25 25 36 49 219 #*,'-9 6'- :# *%%(# -*# #5+$-%%. '?5-$(. xi yi nx 1 3 3 2 2 1 3 3 2 1 3 4 1 1 ny 3 2 3 1 1 n = 10 .*#9 2%#%-# #8#,,-- Y % X. *,'5++ %.*#%%(# ,299( ,-,'#92 (6.23), -9##9 8 10b0 + 22b1 = 45 < : 22b0 + 58b1 = 111: #<+ ,-,'#92, %1*-9 b0 = 1 75 - b1 = 1 25. 5#*'#5%, 2%#%-# #8#,,-- Y % X -9##' -* yx = 1 75 + 1 25x: .*#9 2%#%-# #8#,,-- X % Y. *,'5++ %.*#%%(# ,299( ,-,'#92 (6.25), -9##9 8 10a0 + 45a1 = 22 < : 45a + 58a = 111: 0 1 #<+ ,-,'#92, %1*-9 a0 = ;1 08 - a1 = 0 73. 5#*'#5%, 2%#%-# #8#,,-- X % Y -9##' -* xy = ;1 08 + 0 73y: ' 6.4 ,3#*#5#%-# 3-&% X - Y 3-*-',+ ,5#*2HO#. #5+$-%%. '?5-$# : 190 X {2 {1 0 1 2 my Y 0 10 20 30 40 50 1 2 1 1 3 3 2 4 1 5 3 4 7 5 4 2 4 60 70 80 mx 4 1 8 4 2 12 5 1 12 3 6 5 7 4 42 .'- #5+$-%%(# '%<#%-+ yjx - xjy - ,%-' -1 , ,'#','2HO-9 644-$-#%'9 5-%#.%. #8#,,--. () ,A#' #5+$-%%(1 '%<#%-. ?2*#9 3-&*-' 3 ,5#*2HO#. ,1#9#: P mx 8 P m x2 60 = 1 429U x x 2 X = P m = 42 = 0 190U X = Px mx = 42 x x x x X2 = X2 ; (X)2 = 1 393: & -,1*%(1 *%%(1 ,5#*2#', A' y0 = 40 - = 0,1. ,35&2#9,+ $#%'-%%. #5-A-%. V = (Y { 40)/10. 5+ %#E P m v2 P mv 0 y y y V = P m = 42 = 0 V2 = Py my y = 302 42 = 7 190 y y y y V2 = V2 ; (V)2 = 7 190: 5## P 2 m v xy y 0 = 0 y (Vx)2 = = 42 P m x x 2(Vx) = (Vx)2 ; (V)2 = 0 P 1 (P m x)2 x xy 542 = 1 251 (Xv )2 = y myP m = 5242 P - 1 x mx y ',H* y jx xjy = xjv y 2(Xv ) = (Xv )2 ; (X)2 = 1 215: v s u u 2 Vx u 0 = 0U = vjx = t 2 = 7 190 Y v s u u 2 Xv q u = t 2 = 1 215 = 0 872 = 0 934: 1 393 X A#-*%, A' yjx = xjy = 0 - r(X Y) = 0. 191 ' 6.5 9##',+ *# (?- (x1 x2 : : : xn) - (y1 y2 : : : yn) ?T#99 n :*+. ,'-' ,-,'#92 %95%(1 2%#%-., ,A-'+, A' #5+$-%%+ &-,-9,' 3-&% Y % 3-&% X -9##' -5-%#.%(. 3?5-A#,-. -* yx = a + bx + cx2 (-,. 6.14). () ,%# 9#'* %-9#%<-1 *' 39#'( a b c ?2*#9 3#*#5+' -,1*+ -& 9-%-929 42%$-%5 %#+&n X (yi ; yx)2 ) min : i=1 *,'5++ 6' ,'%<#%-# 3?5-A#,2H &-,-9,' yx = a + bx + cx2, 3,5# *-44##%$-%-+ 3 39#'9 a, b, c 352A-9 -,92H ,-,'#92 ( %#. n *5+ ?O%,'- #5-A-% n ?&%A#% A##& P 1) i=1 8 Pn Pn x + c Pn x2 = Pn y > 1 + b a i i > > i=1 i=1 i=1 i i=1 > > > < Pn Pn x2 + c Pn x3 = Pn y x a x + b i i i > i=1 i=1 i i=1 i i=1 > > > > n n n n > : a P x2i + b P x3i + c P x4i = P yix2i : i=1 i=1 i=1 i=1 %58-A% 9:% 3,'-' ,-,'#92 %95%(1 2%#%-. - '9 ,52A#, 8* 3#*358#9+ #5+$-%%+ &-,-9,' Y % X -9##' -5-%#.%(. 3?5-A#,-. -* ,'#3#%-, ?5## (,. A#9 2. y 6 r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r 6.14 | $ + ' 6.6 r - x & 8#%#5%. ,23%,'-, ,3#*#5#%-# 3-&% X - Y '. %95%#, 3-&#*#% (? ?T#99 530 #*-%-$. #&25''( -&9##%-+ 3-&% X - Y 2 93%#%' (?- 3-*+',+ ,5#*2HO#. '?5-$# : 192 X Y 15{25 25{35 35{45 45{55 55{65 65{75 75{85 my 200{300 19 5 24 300{400 23 116 11 150 400{500 1 41 98 9 149 500{600 4 32 65 7 108 600{700 1 4 21 36 3 65 700{800 1 2 11 13 1 28 800{900 1 3 2 6 mx 43 167 146 97 55 19 3 530 .'- (?A%(. 644-$-#%' 5-%#.%. #5+$-- - #8 ,#*%## *'-A#,# '5%#%-#. 3-,' 2%#%-+ 3+9(1 #8#,,-. Y % X - X % Y. () %A5 -%'#5( &%A#%-. 3-&% X - Y &9#%-9 -1 ,##*-%9-. #&25''# 352A-9 #5+$-%%2H '?5-$2. 644-$-#%' 5-%#.%. #5+$-- %# -&9#%+#',+ ' -&9#%#%-+ %A5 ',A#' - 9,<'? 3-&% (,.,' 2), ' *5+ 23O#%-+ ,A#' &9#%-9 &%A#%-+ 3-&% X - Y % &%A#%-+ 3-&% U - V, ,+&%%(1 , 3#(9,5#*2HO-9- ,'%<#%-+9- : U = (X ; x0) V = (Y ; y0) '(1 35:-9 x0 = 550U y0 = 50U = 0,01U = 0,1. 52A#%%(# &%A#%-+ - &3-<#9 %* ,'#','2HO-9- -%'#59-. ,# *5%#.<-# (A-,5#%-+ 3-&*+',+ %* &%A#%-+9- 3-&% U - V : XX mij uivj = 1161 UV = N1 530 = 2 191U i j X 507 = ;0 957U V = N1 myj vj = ;530 j X 382 = ;0 721: U = N1 mxi ui = ;530 i 5## X V2 = N1 myj vj2 = 1359 530 = 2 564U j A' *#' '2* -9##9 X U2 = N1 mxi u2i = 1196 530 = 2 257 i V2 = V2 ; V2 = 2 564 ; 0 916 = 1 648U U2 = U2 ; U2 = 2 257 ; 0 520 = 1 737 V = 1 28U 193 U = 1 32: 5+ 644-$-#%' #5+$-- 352A-9 ; 0 690 = 1 501 = 0 888: r = UV ; U V = 21191 32 1 28 1 690 U V 5## 721 = 550 ; 72 1 = 477 9U X = x0 + U = 550 + ;0001 Y = y0 + V = 50 + ;0 957 = 50 ; 9 57 = 40 43: 0 1 5+ 644-$-#%' #8#,,-- %.*#9 28=0 1) = 0 888 12 8 = 0 0861U yjx = r Y = 0 888 (1(132 =0 01) 132 X xjy = r X = 9 157: Y 6'92 2%#%-+ 3+9(1 #8#,,-. Y % X - X % Y ,5#*2HO-# : y ; 40 43 = 0 0861 (x ; 477 9)U x ; 477 9 = 9 157 (y ; 40 43) *5+ (?A%8 644-$-#%' 5-%#.%. #5+$-- - #8 ,#*%#8 *'-A#,8 '5%#%-+ 352A-9 : ; r2 = 0 009: r 1 p n ' 6.7 9##',+ (? ?T#9 n = 11, -&5#A#%%+ -& *29#%. %95%. ,23%,'-. 5+ %#E (A-,5#% (?A%(. (693--A#,-.) 644-$-#%' #5+$-r = 0 76. #?2#',+ : ) 3,'-' 95%{. *#-'#5%(. -%'#5 *5+ 644-$-#%' #5+$-(?A%. ,23%,'-U ?) 3#-' %25#2H 8-3'#&2 fH0 : = 0g 3'- 5'#%'-%. 8-3'#&( fH : 6= 0g. #% &%A-9,'- 3-%+' 0,01. () .*#9 95%{. *#-'#5%(. -%'#5 *5+ 644-$-#%' #5+$-- (?A%. ,23%,'- . (35%- ,A#'(, 35&2+, ,3#*#5#%-#9 9#, 352A-9 *5+ 95%{8 *#-'#5%8 -%'#5 +0 27 < < +0 92. 194 *%%(. *#-'#5%(. -%'#5 %# %(#' &%A#%-# = 0, ' % 2%# &%A-9,'- = 0 01 '#8#',+ %25#+ 8-3'#& fH0 : = 0g %##5-%%,'- 3##9#%%(1 X - Y. #2 6'. :# %25#. 8-3'#&( 9:% 3#,'-, -,35&2+ *#5 1. 5+ 6'8 (A-,5-9 ,''-,'-2 s s n ; 2 ; 2 = 3 50: t = r 1 ; r2 = 0 76 1 11 ; 0 762 '?5-$9 ,'%*'-&-%%8 %95%8 ,3#*#5#%-+ 3 = 0 01 %1*-9 u=2 = u0005 = 2 58. 5#*'#5%, 3-9#%++ *#5 3, ':# 3-1*-9 (*2 '9, A' (?A%(. 644-$-#%' #5+$-- '5-A#% ' %25+. '?5-$9 ,3#*#5#%-+ 'H*#%' 3 = 0 01 - A-,52 ,'#3#%#. ,?*( = n ; 2 = 9 %1*-9 t=2 n;2 = t00059 = 3 25. t = 3 50 > 3 25, ' (?A%(. 644-$-#%' #5+$-- &%A-9 '5-A#',+ ' %25+, '.#. 3##9#%%(# X - Y +5+H',+ #5-%%(9-. 5+ 3#- 6'. :# %25#. 8-3'#&( H0, '.#. = 0, 3-9#%-9 *#5 3. 5+ 6'8 (A-,5-9 ,''-,'-2 u 3 4925# !p 1 + r 1 + 0 0 u = 1 1513 lg 1 ; r ; 1 1513 lg 1 ; ; 2(n ; 1) n ; 3: 0 58+ 0 = 0, 352A-9 76 p11 ; 3 = 2 82: u = 1 1513 lg 01 24 %-+ , u=2 = u0005 = 2 58, 3-1*-9 (*2 '9, A' (?A%(. 644-$-#%' #5+$-- '5-A#% ' %25+. ' 6.8 (32, %#'(9 3#*3-+'-#9 39(<5#%%. 3*2$-- (Y) & ,#9 5#' (X) 1'#-&2#',+ ,5#*2HO-9- *%%(9- : X, ! 1 2 3 4 5 6 7 Y, . . 0,5 0,5 1,5 3,5 6,5 10,5 15,5 (%+' &-,-9,' Y ' X 3 3?5# y = ax2 + bx + c. () 5+ %1:*#%-+ 39#' a, b - c 3 %#?1*-9 #<-' ,-,'#92 8 Pn 4 Pn 3 Pn 2 > a xi + b xi + c xi > > i=1 i=1 i=1 > > > < Pn 3 Pn 2 Pn a xi + b xi + c xi > i=1 i=1 i=1 > > > > n n n > : a P x2i + b P xi + c P 1 i=1 i=1 i=1 195 9#'*2 %-9#%<-1 *' n = P x2i yi i=1 n = P xi yi i=1 n = P yi i=1 '. *%%9 ,52A# n n= 7, xi | 6' '#2O-. 8* ,#9-5#'-, yi | ,'#','2HO+ 3*2$-+, P 1 = n. i=1 ,'5+#9 ,-,'#92 *5+ 3#*#5#%-+ 39#' a, b, c: 8 4676 a + 784 b + 140 c = 1372 > > > < 784 a + 140 b + 28 c = 224 > > > : 140 a + 28 b + 7 c = 38 5: R-,5#%%# #<#%-# 6'. ,-,'#9( 9:% -,', %3-9#, 9#'*9 9# -5- 9#'*9 -,5HA#%-+ 72,,. #&25''# %1*-9 a = 0,5U b = {1,5U c = 1,5. ' 6.9 (( 2; =>>? ?) ,,9'-9 %95%# ,3#*#5#%-# ,-,'#9( *21 #5-A-% (X Y). #& 8%-A#%-+ ?O%,'- 9:% 3#*358', A' 99#%'( 3#8 3+* %( %25H, ' A' 35'%,' #+'%,'- -9##' -* 2 " 2 !# x 2 xy y 1 1 fXY (x y) = 2 p1 ; 2 exp ; 2(1 ; 2) 2 ; + 2 1 2 1 2 1 2 1 exp ; 1 x2 + 2 xy + y2 2p (1) 11 20 2D 02 G 8*# D = 2002 ; 211 = 12 22(1 ; 2) | *#'#9-%%' 9'-$( '(1 99#%'U 2 G = 20 11 = 2 12 2 : 11 02 1 2 1 (?# -& n %?5H*#%%(1 3 &%A#%-. (x1 y1) : : : (xn yn) (A-,5-9 99#%'%(# 1'#-,'-- 3#8 - '8 3+* : n X 1 (2a) x = n xi i=1 n X y = n1 yi (2b) i=1 n n X X 1 1 2 2 m20 = s1 = n (xi ; x) = n x2i ; (x)2 (2c) i=1 i=1 n n X X m02 = s22 = n1 (yi ; y)2 = n1 yi2 ; (y)2 (2d) i=1 i=1 n n X X m11 = rs1s2 = n1 (xi ; x) (yi ; y) = n1 xiyi ; x y: (2r) i=1 i=1 644-$-#%' #5+$-r = sms11 : 1 2 196 #3# ,,9'-9 1'#-,'-A#,2H 42%$-H ,3#*#5#%-+ 3+'- ,52A.%(1 #5-A-% x, y, m20, m11, m02, '+ #,' 42%$-+ ' 3+'- 829#%' t1, t2, t20, t11, t02 : h i Q(t1 t2 t20 t11 t02) = M ei(t1x+t2 y+t20 m20 +t11 m11+t02 m02 ) = Z Z = (2)n1Dn=2 : : : e" dx1 : : : dxn dy1 : : :dyn (3) 8*# n X 02x2i ; 211xiyi + 20yi2 + ^ = ; 21D i=1 +i(t1x + t2y + t20m20 + t11m11 + t02m02) -%'#8-%-# ,3,'%+#',+ % 2n-9#%# 3,'%,' 3##9#%%(1 x1, x2 : : : , xn , y1, y2 : : : , yn. 9#%-9 x1 : : : xn %(9- 3##9#%%(91 : : : n , 39OH '8%5%8 p 3#?&%-+, 3- '9 1 = nx, - 3-9#%-9 3#?&%-# , '. :# 9'-$#. #5-A-%9 p y1 : : : yn, '(# 3- 6'9 &9#%+H',+ %(9- #5-A-%91 : : : n , 3-AE9 1 = ny. 8* -9##9 n n n n n n X X X X X X 2 2 2 xi = i xiyi = i i yi = i2 i=1 i=1 n X nm20 = - ',H* i=2 i=1 i2 nm11 = i=1 n X i=2 ii i=1 nm02 = i=1 n X i=2 i2 ^ = i t11p+ t21 ; 1 0212 ; 21111 + 2012 ; n 2D n20 n X n02 2 + 2 ; n11 ; 1 it 2 ; it + ; it ; n1 20 i 11 i i 02 i : 2 M 2 D 2 2 D i=2 *,'5++ 6' (:#%-# *5+ ^ 49252 (3), 3-#*E9 2n-'%(. -%'#85 3-&#*#%-H n *.%(1 -%'#85. 8* ,9#,'%+ 1'#-,'-A#,+ 42%$-+ Q(t1 t2 t20 t11 t02) 3-9#' -* 1 2 2 Q(t1 t2 t20 t11 t02) = exp ; 2n 20t1 + 211t1t2 + 02t2 (A=A)(n;1)=2 (4) 8*# 02 )=2D ;(n11)=2D A = det ;((n n11)=2D (n20)=2D (n )=2m ; it 11)=2D ; it11=2 : A = det ;(n 02)=2D ; it 20=2 ;((n n20)=2D ; it02 11 11 9#,'%+ 1'#-,'-A#,+ 42%$-+ #,' 3-&#*#%-# *21 9%:-'#5#., 3#(. -& '(1 ,*#:-' 5-< 3##9#%%(# t1 - t2, '. | 5-< t20, t11 - t02. #(. 9%:-'#5 +5+#',+ 1'#-,'-A#,. 42%$-#. %#'8 %95%8 ,3#*#5#%-+ , %25#(9 ,#*%-9 &%A#%-#9 - 9'-$#. '(1 99#%' n;1 G. '. 9%:-'#5 #,' A,'%(. ,52A. 1'#-,'-A#,. 42%$-qn (t20 t11 t02) = (A=A)(n;1)=2 : 197 8*# '#','2HO## ,3#*#5#%-# #,' A,'%(. ,52A. ,3#*#5#%-+ (n;1)=2 fn (x11 x12 x22) = C2n a11a22 ; a212 x11x22 ; x212 (n;4)=2 exp Z;a11x11 ; a22x22 ; 2a12x12] (5) n;3 C2n = p n;11 n;2 = ;(2n ; 2) ; 2 ; 2 , &9#%. x11, x12, x22 % m20, m11, m02. -9 ?&9, ,,'%(# ,52A.%(# #5-A-%( (x, y) - (m20, m11, m02) %#&-,-9(. 9#,'%# ,3#*#5#%-# #5-A-% x - y %95% - -9##' '# :# 99#%'( 3#8 3+*, A' - ,3#*#5#%-# 8#%#5%. ,23%,'-, - 9'-$2 '(1 99#%' n;1G. 9#,'%# ,3#*#5#%-# #5-A-% m20, m11, m02 -9##' 35'%,' ,3#*#5#%-+ #+'%,'- fn, &*#92H 4925. n;1 2 (n;4)=2 fn (m20 m11 m02) = 4;(n n ; 2) (m20m02M;(n;m1)11=2) (6) n exp ; 2M (02m20 ; 211m11 + 20m02) ?5,'- m20 > 0, m02 > 0 - m211 < m20m02, 3-A#9 fn = 0 %# 6'. ?5,'-. #*#9 %2H 3##9#%%2H r ,9#,'%# ,3#*#5#%-# (6) #5-A-% m20, m11 p - m02, 358+ m11 = r m20m02, ' A' r | 644-$-#%' #5+$-- (?-. 8* 352A-9 ,5#*2HO## (:#%-# *5+ ,9#,'%. 35'%,'- ,3#*#5#%-+ #+'%,'#. #5-A-% m20, m02 - r: pm m f (m rpm m m ) = 20 02 n 20 20 02 02 (7) n;1 = 4;(n ;n 2)D(n;1)=2 (m20m02)(n;3)=2 (1 ; r2)(n;4)=2 exp ; 2nD (02m20 ; 211rpm20m02 + 20m02) 8*# m20 > 0, m02 > 0, r2 < 1. R,'%+ 35'%,' ,3#*#5#%-+ #+'%,'- *5+ r 352A-',+ -%'#8-%-#9 ,9#,'%. 35'%,'- ,3#*#5#%-+ #+'%,'-3 m20 -pm02 3#*#51 ' 0 * 1. ,5- &5:-' ,'#3#%%. +* 9%:-'#5 exp ;n11r m20m02=D , ' 9:% 352A-' 35'%,' ,3#*#5#%-+ #+'%,'#. *5+ (?A%8 644-$-#%' #5+$-- r : fn (r) = (8) ) 1 n;3 X = (n2 ; 3)! (1 ; 2)(n;1)=2(1 ; r2)(n;4)=2 ;2 n + 2 ; 1 (2r ! =0 3- ;1 < r < 1. 198 1*+O-. 6' (:#%-# ,'#3#%%. +* 9:% 3#?&'. ,52 ,3#*5- n + ; 1 (2r) Z1 xn;2 1 2n;3 X dx 2 p ; = (9) n;1 1 ; x2 (n ; 2)! =0 2 ! (1 ; rx ) 0 ' ',H* 352A#',+ ,5#*2HO## (:#%-# *5+ 35'%,'- ,3#*#5#%-+ #+'%,'#. fn (r) #5-A-%( r : Z1 xn;2 dx : n ; 2 2 ( n ; 1) = 2 2 ( n ; 4) = 2 p (1 ; r ) (10) fn(r) = (1 ; ) n ; 1 (1 ; rx) 1 ; x2 0 .*#%%# ,3#*#5#%-# &-,-' '5 ' ?T#9 (?- n - ' 644-$-#%' #5+$-- 8#%#5%. ,23%,'- . - n = 2 35'%,' ,3#*#5#%-# #+'%,'#. fn (r) ?O#',+ %25 ,'#','-- , '#9 4'9, A' 644-$-#%' #5+$--, (A-,5#%%(. 3 (?#, ,,'+O#. '5 -& *21 &%A#%-., %#?1*-9 #% 1, ' A' 6'9 ,52A# ,3#*#5#%-# 3-%*5#:-' *-,#'%92 '-32. - n = 3 35'%,' ,3#*#5#%-# #+'%,'#. U-?&%, , ?#,%#A%(9*-%'9- 'A1 r = 1. - n = 4 352A#',+ 3+9285%# ,3#*#5#%-#, #,5- = 0 - J-?&%# ,3#*#5#%-# ,52A# 6= 0. - n > 4 ,3#*#5#%-# #+'%,'#. 2%-9*5%, ## 9* ,,#*'A#% 'A# r = 0, #,5- = 0, - 5 'A- r = , #,5- 6= 0. 6.9. ) * 6.1 2,' &-,-9,' 3-&% Y ' 3-&% X 1'#-&2#',+ ,5#*2HO#. '?5-$#. : X {2 {1 0 1 2 3 Y {2 {3 {3 {1 3 7 #*358+, A' y = ax + b, %.'- 39#'( 6'. &-,-9,'-, 35&2+, 9#'*9 %-9#%<-1 *'. 6.2 2,' &-,-9,' 3-&% Y ' 3-&% X 1'#-&2#',+ ,5#*2HO#. '?5-$#. : X {2 {1 0 1 2 3 Y 2 3 3 1 {3 {7 #*358+, A' y = ax + b, %.'- 39#'( 6'. &-,-9,'-, 35&2+, 9#'*9 %-9#%<-1 *'. 199 6.3 5#*2HO+ #5+$-%%+ '?5-$ *#' ,3#*#5#%-# *21 3-&% X - Y : X 5 6 7 8 10 2 1 20 1 2 30 1 1 2 Y 40 1 2 50 2 1 mx 4 4 5 1 14 my 3 3 3 3 ,'-' &-,-9,'- 5-%#.%. #8#,,-- X % Y - Y % X. 6.4 *#,+'- 8*1 -9#H',+ 3'#-. 5-A#,' 3'# - A-,5#%%,' %,#5#%-+ ( *#,+'1 '(,+A A#5#) 6'-1 10 8*1 3-#*#% '?5-$#. %#,'- *%%(# % *-8992 ,,#+%-+, (A-,5-' 644-$-#%'( #5+$-- : ) *5+ 3#(1 *#+'- 8*, ?) *5+ ,#1 *#,+'- 8*. %-' #&25''(. } 8* 1 2 3 4 5 6 7 8 9 10 ,#5#%-# 45 45 47 48 51 58 59 65 67 802 R-,5 3'# 12 12 29 25 38 35 16 43 22 345 6.5 (A-,5-' 644-$-#%' 5-%#.%. #5+$-- - 352A-' 2%#%-# 3+9. #8#,,-- X % Y 3 *%%(9 '?5-$( ,3#*#5#%-+ 100 ,'#%-. :- 3 ?O#92 #,2 ,#8 ,'#%-+ Y - 3 #,2 ,#9+% X : X, Y, 20- 30- 40- 50- 60- 70- 80- 90- 100- 110- 120-30 -40 -50 -60 -70 -80 -90 -100 -110 -120 -130 10{15 3 15{20 2 6 1 1 20{25 4 13 2 1 25{30 5 4 30{35 8 4 2 35{40 1 4 6 40{45 2 6 1 45{50 1 1 5 1 50{55 4 2 55{60 1 4 1 60{65 1 65{70 1 1 1 6.6 #*%++ '#93#'2 &*21 ,#%'+?# *21 8*1 (X) - (Y) -&9#+5, '#A#%-# 40 5#'. %%(# 3-#*#%( '?5-$#. 200 X 12,0 12,0 12,0 12,0 12,8 13,8 13,1 13,0 Y 10,8 11,3 12,0 13,0 10,9 10,0 13,0 13,0 X 13,9 14,2 14,0 14,0 13,9 15,0 14,0 14,0 Y 10,1 10,0 10,0 12,0 12,4 11,0 14,8 15,2 X 14,9 14,9 15,1 15,0 15,5 15,9 16,0 15,9 Y 13,0 14,2 13,8 16,0 13,9 14,7 13,0 15,0 X 16,0 16,9 17,2 16,9 16,9 17,0 16,8 17,5 Y 16,0 12,9 13,9 14,8 15,0 16,0 17,0 16,0 X 18,0 18,0 18,1 18,4 19,2 19,3 20,0 20,1 Y 14,0 14,9 16,0 17,8 15,0 16,1 17,0 17,7 .'- (?A%(# ,#*%#9#,+A%(# '#93#'2( ?-1 %,#5#%%(1 32%'1 - -1 ,#*%#*'-A%(# 25%#%-+. .'- (?A%(. 644-$-#%' #5+$-X - Y, %3-,' (?A%# 2%#%-# 5-%#.%. #8#,,-- Y % X. 6.7 & 125 3('%(1 2A,' 55 %1*-5-, % %#2*?#%%9 9,,-# - 70 % 2*?#%%9. - 6'9 % ,#1 125 2A,'1 -9#5 9#,' ,5#*2HO+ 2:.%,' : #*%-. 2:., !/2 95 105 115 125 135 145 155 165 175 R-,5 2A,' % I 9,,-# 2 5 12 15 10 7 3 1 0 R-,5 2A,' % II 9,,-# 0 0 1 2 8 24 19 11 5 .'- #5+$-%%# '%<#%-# 2:.%,'- ' 2*?#%-.. 6.8 -:# 3-#*#% '?5-$ &%A#%-. 3-&% Y 3- &5-A%(1 &%A#%-+1 3-&% X : X 12 13 14 16 17 18 20 21 21 24 25 26 28 Y 54 59 67 76 85 97 107 118 127 139 153 160 178 (%+' &-,-9,' Y ' X *5 3+9. y = ax + b, 35&2+, 9#'*9 %-9#%<-1 *'. (%+' &-,-9,' X ' Y *5 3+9. x = cy + d, 35&2+, 9#'*9 %-9#%<-1 *'. 52A#%%(# &-,-9,'- ,%-'. 6.9 2,' &-,-9,' 3-&% Y ' 3-&% X 1'#-&2#',+ ,5#*2HO#. '?5-$#. : X {2 {1 0 1 2 3 Y {2 {3 {3 {1 3 7 #*358+, A' y = ax2 +bx+c, %.'- 39#'( 6'. &-,-9,'-, 35&2+, 9#'*9 %-9#%<-1 *'. 6.10 3('1 ?(5- 352A#%( ,5#*2HO-# #&25''( 3- -&9##%-+1 *-9#' 201 (Y) 3(5$( <-*%. 42,-- &-,-9,'- ' A-,5 ,3 (X), ,35:#%%(1 6'-5%. 35,,'- 3(5-%- : X 10 15 20 25 30 35 40 45 50 55 60 0 3 7 1 3 6 6 Y, 2 3 1 4 5 1 3 4 3 4 3 4 3 my 3 10 6 7 4 5 4 4 6 4 3 .'- 644-$-#%' 5-%#.%. #5+$-- 9#:*2 3-&%9- X - Y 2%#%-# 5-%#.%. #8#,,-- y % x. 6.11 -,-9,' 3*5:-'#5%,'- t #<#%-+ ,-,'#9 5-%#.%(1 2%#%-. *-%. ,'#3#%- '2*%,'- ' 3+* ,-,'#9( n 3-#*#% ,5#*2HO#. '?5-$# : n 2 3 4 5 6 7 8 9 10 t, 12 35 75 130 210 315 445 600 800 #*358+ ,3#*5-,' '8, A' t = An, %.'- &%A#%-+ 39#' A - 9#'*9 %-9#%<-1 *'. 6.12 ,#9 & 3- &5-A%(1 &%A#%-+1 3-&% X ?(5 -&9##% &%A#%-# 3-&% Y. 52A#%%(# #&25''( 3-*+',+ ,5#*2HO#. '?5-$# : X 0,30 0,91 1,50 2,00 2,20 2,62 3,00 3,30 Y 0,20 0,43 0,31 0,52 0,81 0,68 1,15 0,85 #*358+ '##'-A#,-, A' y = ax + b, %.'- a - b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b0 - b1 5-%#.%8 2%#%-+ #8#,,-- Y % X -* yx = b0 + b1x 3 9#'*2 %-9#%<-1 *'. 15. 3-<-'# ,-,'#92 2%#%-. *5+ 3#*#5#%-+ 644-$-#%' a0 - a1 5-%#.%8 2%#%-+ #8#,,-- X % Y -* xy = a0 + a1y 3 9#'*2 %-9#%<-1 *'. 16. A#9 ,,'-' ,.,' 9-%-95%,'- 9*#5%(1 42%$-. #8#,,--? 17. 3,''# * 2%#%-+ 5-%#.%. #8#,,--: Y % X - X % Y. +,%-'# ,+& 9#:*2 644-$-#%'9- a0, a1 - b0, b1. 203 #$ .1. ! %,'+O#9 3-5:#%-- 3,?-H 3-#*#%( '?5-$( #5-A-%, A,' -,35&2#9(1 3- #<#%-- &*A 9'#9'-A#,. ,''-,'-- : 1. 95%+ 42%$-+ 72,, exp(;x2), 3-&*%+ %95%. 35'%,'42%$-+ <-? erf(x) - *35%-'#5%+ 42%$-+ <-? erfc(x)U 2. J2%$-+ K35, a(x) - 42%$-- da(x)=dx, 2a(x), 1 ; 2a(x)U 3. %'-5- u %95%8 ,3#*#5#%-+ 72,,, '#AHO-# #%,'2 = 1 ; a(u)U 4. 7%-$( 1 - 2 *#-'#5%8 -%'#5 *5+ %95%. ,52A.%. #5-A-%(: p = Pr (1s < < 2s) = 1 ; U 2 2 5. %'-5 2 2 ,3#*#5#%-+ , ,'#3#%+9- ,?*(, '#AHO-# #%,'2 Pr > = U 2 2 6. ,'%%-# %'-5 2 2 1; ,3#*#5#%-+ , ,'#3#%+9- ,?*(, '#AHO-# #%,'2 Pr > 1; = 1 ; U 7. %'-5- t ,3#*#5#%-+ 'H*#%' , ,'#3#%+9- ,?*(, 3#*#5+#9(# #+'%,'H Pr (jtj > t ) = (*2,'%%++ -'-A#,+ ?5,')U 8. K#,'%%-# %'-5- ,3#*#5#%-+ 'H*#%' tp(k), , k ,'#3#%+9,?*(, '#AHO-# #%,'2 Pr (t < tp(k)) = pU 9. %'-5- ,3#*#5#%-+ F k1 k2 J-<#-%#*# , A-,59- ,'#3#%#. ,?*( k1 - k2U 10. #*#5%# ,3#*#5#%-# 598 K ()U 11. %'-5- ,3#*#5#%-+ 598: Pr ( ) = U 12. %'-5- r A-,5 &%: Pr (r r) = . (2);1=2 exp(;x2=2), 204 exp(;x ), (2) = exp(;x =2), !" erf(x) # !" erfc(x) 1. 2 ;1 2 x exp(;x2) (2);1=2 exp(;x2=2) 0,00 1,00000 0,39894 0,10 0,99005 0,39705 0,20 0,96079 0,39104 0,30 0,91393 0,38149 0,40 0,85214 0,36837 0,50 0,77880 0,35217 0,60 0,69768 0,33322 0,70 0,61263 0,31235 0,80 0,52729 0,28979 0,90 0,44486 0,26619 1,00 0,36788 0,24207 1,10 0,29820 0,21795 1,20 0,23693 0,19429 1,30 0,18452 0,17147 1,40 0,14086 0,14973 1,50 0,10540 0,12952 1,60 0,07730 0,11092 1,70 0,05558 0,09405 1,80 0,03916 0,07905 1,90 0,02705 0,06562 2,00 0,01832 0,05409 2,10 0,01216 0,04408 2,20 0,00791 0,03557 2,30 0,00504 0,02833 2,40 0,00315 0,02249 2,50 0,00193 0,01753 *#, Zx 2 erf(x) = p exp(;t2) dt 0 205 erf(x) 0,00000 0,11246 0,22270 0,32863 0,42839 0,52050 0,60386 0,67780 0,74210 0,79691 0,84270 0,88021 0,91031 0,93401 0,95229 0,96611 0,97635 0,98379 0,98909 0,99279 0,99532 0,99702 0,99814 0,99886 0,99931 0,99959 erfc(x) 1,00000 0,88754 0,77730 0,67137 0,57161 0,47950 0,39614 0,32220 0,25790 0,20309 0,15730 0,11979 0,08969 0,06599 0,04771 0,03389 0,02365 0,01621 0,01091 0,00721 0,00468 0,00298 0,00186 0,00114 0,00069 0,00041 erfc(x) = 1 ; erf(x): 2 $ %(x) &# # 2. x 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9 2,0 2,1 2,2 2,3 2,4 2,5 *#, da(x)=dx 0,3989 0,3950 0,3833 0,3646 0,3400 0,3107 0,2783 0,2444 0,2104 0,1775 0,1468 0,1190 0,0945 0,0736 0,0562 0,0420 0,0308 0,0222 0,0156 0,0108 0,0073 0,0048 0,0032 0,0020 0,0013 0,0008 a(x) 0,0000 0,0398 0,0793 0,1179 0,1554 0,1915 0,2257 0,2580 0,2881 0,3159 0,3413 0,3643 0,3849 0,4032 0,4192 0,4332 0,4452 0,4554 0,4641 0,4713 0,4772 0,4821 0,4861 0,4893 0,4918 0,4938 2 a(x) 0,0000 0,0797 0,1585 0,2358 0,3108 0,3829 0,4515 0,5161 0,5763 0,6319 0,6827 0,7287 0,7699 0,8064 0,8385 0,8664 0,8904 0,9109 0,9281 0,9426 0,9545 0,9643 0,9722 0,9786 0,9836 0,9876 1 ; 2 a(x) 1,0000 0,9203 0,8415 0,7642 0,6892 0,6171 0,5485 0,4839 0,4237 0,3681 0,3173 0,2713 0,2301 0,1936 0,1615 0,1336 0,1096 0,0891 0,0719 0,0574 0,0455 0,0357 0,0278 0,0214 0,0164 0,0124 Zx 1 exp(;t2=2) dt: a(x) = p 2 0 p p a(x) = 21 erf(x= 2) erf(x) = 2 a( 2 x): 206 3. ' u ( ### - *%%9 2%# &%A-9,'- 3,'%%-. %'-5 u 3#*#5+#',+ #%,'9 : Z1 1 = p u exp ; t2=2 dt = 21 ; a(u) : 2 #% &%A-9,'- 0,0001 0,005 0,010 0,015 0,020 0,025 0,030 0,035 0,040 0,045 0,050 %'-5 u 3,0902 2,5758 2,3263 2,1701 2,0537 1,9600 1,8808 1,8119 1,7507 1,6954 1,6449 - A-,5#%%9 %1:*#%-- &%A#%-. 42%$-- K35, a(x) - 3,'%%#8 %'-5+ u 9:% ':# 35&',+ ,5#*2HO-9- 3-?5-:#%%(9- (:#%-+9- : Z1 2 p qx = 2Z1 ; a(x)] = exp ;t2=2 dt = 2 x ! (83 x + 35) x + 562 = exp ; 165 + 703=x 0 < x 5 5U " #;1=2 ((4 y + 100) y + 205) y x = ((2y + 50)y + 192)y + 13 2 10;7 qx 1 8*# y = ; ln qx. 207 )*# #+# (& ##( #, #,-.# # p = Pr ( s < < s) = 1 ; 4. 1 2 1 s 2 4 1 = n ; 1 (xi ; i n X =1 3= x) 5 1 2 2 x 2 n = n1 X xi i=1 C p = 1 ; = 0 99 p = 1 ; = 0 95 p = 1 ; = 0 90 1 2 1 2 1 2 1 0,356 159 0,446 31,9 0,510 15,9 2 0,434 14,1 0,521 6,28 0,578 4,40 3 0,483 6,47 0,566 3,73 0,620 2,92 4 0,519 4,39 0,599 2,87 0,649 2,37 5 0,546 3,48 0,624 2,45 0,672 2,090 6 0,569 2,98 0,644 2,202 0,690 1,916 7 0,588 2,66 0,661 2,035 0,705 1,797 8 0,604 2,440 0,675 1,916 0,718 1,711 9 0,618 2,277 0,688 1,826 0,729 1,645 10 0,630 2,154 0,699 1,755 0,739 1,593 11 0,641 2,056 0,708 1,698 0,748 1,550 12 0,651 1,976 0,717 1,651 0,755 1,515 13 0,660 1,910 0,725 1,611 0,762 1,485 14 0,669 1,854 0,732 1,577 0,769 1,460 15 0,676 1,806 0,739 1,548 0,775 1,437 16 0,683 1,764 0,745 1,522 0,780 1,418 17 0,690 1,727 0,750 1,499 0,785 1,400 18 0,696 1,695 0,756 1,479 0,790 1,385 19 0,702 1,666 0,760 1,460 0,794 1,370 20 0,707 1,640 0,765 1,444 0,798 1,358 21 0,712 1,617 0,769 1,429 0,802 1,346 22 0,717 1,595 0,773 1,416 0,805 1,335 23 0,722 1,576 0,777 1,402 0,809 1,326 24 0,726 1,558 0,781 1,391 0,812 1,316 25 0,730 1,541 0,784 1,380 0,815 1,308 26 0,734 1,526 0,788 1,371 0,818 1,300 27 0,737 1,512 0,791 1,361 0,820 1,293 28 0,741 1,499 0,794 1,352 0,823 1,286 29 0,744 1,487 0,796 1,344 0,825 1,279 30 0,748 1,475 0,799 1,337 0,828 1,274 208 5. ',## , ### , #,-.# # Pr > = 2 2 2 R-,5 ,'#3#%#. ,?*( 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0,01 6,6 9,2 11,3 13,3 15,1 16,8 18,5 20,1 21,7 23,2 24,7 26,2 27,7 29,1 30,6 32,0 33,4 34,8 36,2 37,6 38,9 40,3 41,6 43,0 44,3 45,6 47,0 48,3 49,6 50,9 #% &%A-9,'- (*2,'%%++ -'-A#,+ ?5,') 0,025 0,05 0,95 0,975 0,99 5,0 3,8 0,0039 0,001 0,00016 7,4 6,0 0,103 0,051 0,020 9,4 7,8 0,352 0,216 0,115 11,1 9,5 0,711 0,484 0,297 12,8 11,1 1,15 0,831 0,554 14,4 12,6 1,64 1,24 0,872 16,0 14,1 2,17 1,69 1,24 17,5 15,5 2,73 2,18 1,65 19,0 16,9 3,33 2,70 2,09 20,5 18,3 3,94 3,25 2,56 21,9 19,7 4,57 3,82 3,05 23,3 21,0 5,23 4,40 3,57 24,7 22,4 5,89 5,01 4,11 26,1 23,7 6,57 5,63 4,66 27,5 25,0 7,26 6,26 5,23 28,8 26,3 7,96 6,91 5,81 30,2 27,6 8,67 7,56 6,41 31,5 28,9 9,39 8,23 7,01 32,9 30,1 10,1 8,91 7,63 34,2 31,4 10,9 9,59 8,26 35,5 32,7 11,6 10,3 8,90 36,8 33,9 12,3 11,0 9,54 38,1 35,2 13,1 11,7 10,2 39,4 36,4 13,8 12,4 10,9 40,6 37,7 14,6 13,1 11,5 41,9 38,9 15,4 13,8 12,2 43,2 40,1 16,2 14,6 12,9 44,5 41,3 16,9 15,3 13,6 45,7 42,6 17,7 16,0 14,3 47,0 43,8 18,5 16,8 15,0 209 2 0# ## "&, #,-.# # Pr > = 1 ; 6. 2 1; 2 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0,70 1,07 2,41 3,67 4,88 6,06 7,23 8,38 9,52 10,7 11,8 12,9 14,0 15,1 16,2 17,3 18,4 19,5 20,6 21,7 22,8 23,9 24,9 26,0 27,1 28,2 29,2 30,3 31,4 32,5 33,5 0,80 1,64 3,22 4,64 5,99 7,29 8,56 9,80 11,0 12,2 13,4 14,6 15,8 17,0 18,2 19,3 20,5 21,6 22,8 23,9 25,0 26,9 27,3 28,4 29,6 30,7 31,8 32,9 34,0 35,1 36,3 0,90 2,71 4,61 6,25 7,78 9,24 10,6 12,0 13,4 14,7 16,0 17,3 18,5 19,8 21,1 22,3 23,5 24,8 26,0 27,2 28,4 29,6 30,8 32,0 33,2 34,4 35,6 36,7 37,9 39,1 40,3 2 1; 1; 0,95 0,975 0,990 0,995 0,999 3,84 5,02 6,63 7,88 10,8 5,99 7,38 9,21 10,6 13,8 7,81 9,35 11,3 12,8 16,3 9,49 11,1 13,3 14,9 18,5 11,1 12,8 15,1 16,7 20,5 12,6 14,4 16,8 18,5 22,5 14,1 16,0 18,5 20,3 24,3 15,5 17,5 20,1 22,0 26,1 16,9 19,0 21,7 23,6 27,9 18,3 20,5 23,2 25,2 29,6 19,7 21,9 24,7 26,8 31,3 21,0 23,3 26,2 28,3 32,9 22,4 24,7 27,7 29,8 34,5 23,7 26,1 29,1 31,3 36,1 25,0 27,5 30,6 32,8 37,7 26,3 28,8 32,0 34,3 39,3 27,6 30,2 33,4 35,7 40,8 28,9 31,5 34,8 37,2 42,3 30,1 32,9 36,2 38,6 43,8 31,4 34,2 37,6 40,0 45,3 32,7 35,5 38,9 41,4 46,8 33,9 36,8 40,3 42,8 48,3 35,2 38,1 41,6 44,2 49,7 36,4 39,4 43,0 45,6 51,2 37,7 40,6 44,3 46,9 52,6 38,9 41,9 45,6 48,3 54,1 40,1 43,2 47,0 49,6 55,5 41,3 44,5 48,3 51,0 56,9 42,6 45,7 49,6 52,3 58,3 43,8 47,0 50,9 53,7 59,7 210 7. ' t ### 1-# ## "&, ###&# #- Pr (jtj > t ) = ( ,# ") R-,5 ,'#3#%#. ,?*( 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 1 0,01 2,92 2,35 2,13 2,01 1,94 1,89 1,86 1,83 1,81 1,80 1,78 1,77 1,76 1,75 1,75 1,74 1,73 1,73 1,73 1,72 1,72 1,71 1,71 1,71 1,71 1,71 1,70 1,70 1,70 1,68 1,64 #% &%A-9,'- 0,025 0,05 0,95 0,975 4,30 6,97 9,92 22,33 3,18 4,54 5,84 10,22 2,78 3,75 4,60 7,17 2,57 3,37 4,03 5,89 2,45 3,14 3,71 5,21 2,36 3,00 3,50 4,79 2,31 2,90 3,36 4,50 2,26 2,82 3,25 4,30 2,23 2,76 3,17 4,14 2,20 2,72 3,11 4,03 2,18 2,68 3,05 3,93 2,16 2,65 3,01 3,85 2,14 2,62 2,98 3,79 2,13 2,60 2,95 3,73 2,12 2,58 2,92 3,69 2,11 2,57 2,90 3,65 2,10 2,55 2,88 3,61 2,09 2,54 2,86 3,58 2,09 2,53 2,85 3,55 2,08 2,52 2,83 3,53 2,07 2,51 2,82 3,51 2,07 2,50 2,81 3,49 2,06 2,49 2,80 3,47 2,06 2,49 2,79 3,45 2,06 2,48 2,78 3,44 2,05 2,47 2,77 3,42 2,05 2,46 2,76 3,40 2,05 2,46 2,76 3,40 2,04 2,46 2,75 3,39 2,02 2,42 2,70 3,31 1,96 2,33 2,58 3,09 211 0,99 31,60 12,90 8,61 6,86 5,96 5,40 5,04 4,78 4,59 4,44 4,32 4,22 4,14 4,07 4,01 3,96 3,92 3,88 3,85 3,82 3,79 3,77 3,74 3,72 3,71 3,69 3,66 3,66 3,65 3,55 3,29 $## ### 1-# tp(k) k ## "&, #,-.# # Pr (t < tp(k)) = p 8. k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 1 0,750 1,000 0,816 0,765 0,741 0,727 0,718 0,711 0,706 0,703 0,700 0,697 0,695 0,694 0,692 0,691 0,690 0,689 0,688 0,688 0,687 0,686 0,686 0,685 0,685 0,684 0,684 0,684 0,683 0,683 0,683 0,681 0,679 0,677 0,674 0,900 3,078 1,886 1,638 1,533 1,476 1,440 1,415 1,397 1,383 1,372 1,363 1,356 1,350 1,345 1,341 1,337 1,333 1,330 1,328 1,325 1,323 1,321 1,319 1,318 1,316 1,315 1,314 1,313 1,311 1,310 1,303 1,296 1,289 1,282 0,950 6,314 2,920 2,353 2,132 2,015 1,943 1,895 1,860 1,833 1,812 1,796 1,782 1,771 1,761 1,753 1,746 1,740 1,734 1,729 1,725 1,721 1,717 1,714 1,711 1,708 1,706 1,703 1,701 1,699 1,697 1,684 1,671 1,658 1,645 p 0,975 0,990 0,995 12,706 31,821 63,657 4,303 6,965 9,925 3,182 4,541 5,841 2,776 3,747 4,604 2,571 3,365 4,032 2,447 3,143 3,707 2,365 2,998 3,499 2,306 2,896 3,355 2,262 2,821 3,250 2,228 2,764 3,169 2,201 2,718 3,106 2,179 2,681 3,055 2,160 2,650 3,012 2,145 2,624 2,977 2,131 2,602 2,947 2,120 2,583 2,921 2,110 2,567 2,898 2,101 2,552 2,878 2,093 2,539 2,861 2,086 2,528 2,845 2,080 2,518 2,831 2,074 2,508 2,819 2,069 2,500 2,807 2,064 2,492 2,797 2,060 2,485 2,787 2,056 2,479 2,779 2,052 2,473 2,771 2,048 2,467 2,763 2,045 2,462 2,756 2,042 2,457 2,750 2,021 2,423 2,704 2,000 2,390 2,660 1,980 2,358 2,617 1,960 2,326 2,576 212 0,999 318 22,3 10,2 7,173 5,893 5,208 4,785 4,501 4,297 4,144 4,025 3,930 3,852 3,787 3,733 3,686 3,646 3,610 3,579 3,552 3,527 3,505 3,485 3,467 3,450 3,435 3,421 3,408 3,390 3,385 3,307 3,232 3,160 3,090 9. ' ### F k k !#-1##U k | , ### "& "!# #U k | , ### "& #!# # 1 2 1 2 = 0,05 k1 k2 1 2 3 4 5 161 200 216 225 230 18,51 19,00 19,16 19,25 19,30 10,13 9,55 9,28 9,12 9,01 7,71 6,94 6,59 6,39 6,26 6,61 5,79 5,41 5,19 5,05 5,99 5,14 4,76 4,53 4,39 5,59 4,74 4,35 4,12 3,97 5,32 4,46 4,07 3,84 3,69 5,12 4,26 3,86 3,63 3,48 4,96 4,10 3,71 3,48 3,33 4,84 3,98 3,59 3,36 3,20 4,75 3,88 3,49 3,26 3,11 4,67 3,80 3,41 3,18 3,02 4,60 3,74 3,34 3,11 2,96 4,54 3,68 3,29 3,06 2,90 4,49 3,63 3,24 3,01 2,85 = 0,05 6 234 19,33 8,94 6,16 4,95 4,28 3,87 3,58 3,37 3,22 3,09 3,00 2,92 2,85 2,79 2,74 k2 7 237 19,36 8,88 6,09 4,88 4,21 3,79 3,50 3,29 3,14 3,01 2,92 2,84 2,77 2,70 2,66 12 244 19,41 8,74 5,91 4,68 4,00 3,57 3,28 3,07 2,91 2,79 2,69 2,60 2,53 2,48 2,42 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 8 239 19,37 8,84 6,04 4,82 4,15 3,73 3,44 3,23 3,07 2,95 2,85 2,77 2,70 2,64 2,59 9 241 19,38 8,81 6,00 4,78 4,10 3,68 3,39 3,18 3,02 2,90 2,80 2,72 2,65 2,59 2,54 k1 213 10 242 19,39 8,78 5,96 4,74 4,06 3,63 3,34 3,13 2,97 2,86 2,76 2,67 2,60 2,55 2,49 11 243 19,40 8,76 5,93 4,70 4,03 3,60 3,31 3,10 2,94 2,82 2,72 2,63 2,56 2,51 2,45 K () 0,30 0,35 0,40 0,45 0,50 0,55 0,60 0,65 0,70 0,75 0,80 0,85 10. 0### #### '( = P K () 0,0000 0,0003 0,0028 0,0126 0,0361 0,0772 0,1357 0,2080 0,2888 0,3728 0,4559 0,5347 11. 1 (;1) exp ;2 2 2 =;1 0,90 0,95 1,00 1,05 1,10 1,15 1,20 1,25 1,30 1,35 1,40 1,45 K () 0,6073 0,6725 0,7300 0,7798 0,8223 0,8580 0,8878 0,9121 0,9319 0,9478 0,9603 0,9702 1,50 1,55 1,60 1,65 1,70 1,75 1,80 1,85 1,90 1,95 2,00 2,05 K () 0,9778 0,9836 0,9880 0,9914 0,9938 0,9956 0,9969 0,9979 0,9985 0,9990 0,99933 0,99999 ' ### '( Pr ( ) = #% &%A-9,'- 0,20 0,10 0,05 0,02 0,01 0,001 1,703 1,224 1,358 1,520 1,627 1,950 214 12. ' r , Pr (r r) = n #% &%A-9,'- 0,01 0,05 0,10 6 0 0 7 0 0 8 0 0 1 9 0 1 1 10 0 1 1 11 0 1 2 12 1 2 2 13 1 2 2 14 1 2 3 15 2 3 3 16 2 3 4 17 2 4 4 18 3 4 4 19 3 4 5 20 3 5 5 n #% &%A-9,'- 0,01 0,05 0,10 21 4 5 6 22 4 5 6 23 4 6 7 23 4 6 7 24 5 6 7 25 5 7 7 26 6 7 8 27 6 7 8 28 6 8 9 29 7 8 9 30 7 9 10 31 7 9 10 32 8 9 10 33 8 10 11 34 8 10 11 215 % 1. #%'$#5 .. . | .: 2, 1964. 2. 7%#*#% .. 5 . | .: 2, 1961. 3. 5#% .., J-5-33 .. . | .: (,<+ <5, 1973. 4. 28A# .. . | .: 2, 1979. 5. 7-19% .., 1* .., *#% .. . | -#: -O. <., 1979. 6. A#% 7.., #*#*# .. : . | .: (,<. <., 1984. 7. R#'(-% .., 5-19% .K. ) . | .: J-%%,( - ,''-,'-, 1982. 8. 7929% .. . | .: (,<. <., 2000. 9. 9#5+% 7.., -'-A .. 2 . | K.: &*{ K7, 1967. 10. 7#,-9-A .., '## .. . | -%,: (<6.<+ <., 1978. 11. 7929% .. 6 3 $ . | .: (,<. <., 2000. 12. 2? .., #,'+% .., R-,'+ .. ' . | .: 2, 1989. 13. ?%- &*A 3 9'#9'-#. #-+ #+'%,'#. - 9'#9'-A#,+ ,''-,'- = * #*. .. 4-9. | .: 2, 1990. 14. -% .., '#3% . . , +) . | -%,: "%-#,-'#',#", 1987. 15. 2A-% .. % i %: / i . | .: , 1993. 16. &9%-<-5- .. : . | : 7, 1994. 17. - .., -. STATISTIKA. | ' % Windows. | .: "J-5-%", 1997. 216 18. 5H .., '#% .., 1* .., 2?-% .J. ' . | .: 2, 1985. 19. % 7., % . ' . | .: 2, 1983. 20. ?9-A ., '-8% . ' # #. | .: 2, 1979. 21. 5<# K.., 9-% .. # . | .: 2, 1983. @ 4 & 22. R#?(<# .K. . | .{K.: &*{ , 1936. 23. H8# . . | .: 2, 1972. 24. 9# 7. . | .: -, 1975. 25. .. . ># . , !. | .: 2, 1984. 26. J#55# . ) $ . | .: -, 1984. { .1U 1984. { .2. 27. -'35,-. .. % % . | .: 2, 1971. 28. 9-% .., 2%-%{ ,-. .. 5 % % 1 . | .: 2, 1969. 29. %<- K. . | .: -, 1968. 30. %8 . & { 1 . | .: 2, 1965. 31. -% .7., -25-% .. /; # % . | .: 2, 1989. 32. &% .. , # . | .: 2, 1985. 33. '(% 7.. 5 !{. | .: 2, 1978. 34. _#44# 7. < . | .: 2, 1980. 35. , ., -%5- . . | .: -, 1984. 36. -#5 ., ,9 . . | .: J-%%,( ,''-,'-, 1983. 37. % *# *#% .K. . | .: &*{ -%,'. 5-'., 1960. 38. #?# :. 4 ! . | .: -, 1982. 39. H?# . 6 . | .: -, 1984. 40. 5-9 .. & . | .: 2, 1971. 41. ##. 7. , . | .: -, 1990. 42. -5, .. . | .: 2, 1967. 217 A5%# -*%%+ I_IKI 5#,%* #8i.-A R A5%-. 3,i?%- * 3'-A%-1 &%+' ,i.,H 9H ?'2 * -*%%+ #9#%*2 .. 7i5-. #*' .. 9i%i% $ 2002, . 68 $i. 20.10.2002 J 6094 1/16 > { +#i. 8i Times New Roman. Q.{. . 15,0. ( 200 . \. ] $i #. . . . 12,0. î +i , () "*$I", 61002, *i, . J, 21 'i _i1 >: ] 116 i 10.07.2000 > () "*$I", 61002, *i, . J, 21

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