Элементы математической статистики. Учебное пособие
.pdf$9L9A<9
#BEFDB<@ EF4F<EF<K9E><= DS8 BFABE<F9?PAOI K4EFBF
|
X(I) |
|
0 |
|
|
|
|
1 |
|
|
|
2 |
|
|
|
|
3 |
|
|
|
|
|
4 |
|
|
|
5 |
|
|
|
6 |
|
|
|
7 |
|
|||||||||||||||||
p ú ý |
ni |
|
|
4 |
|
|
|
|
13 |
|
|
14 |
|
|
|
|
24 |
|
|
|
|
|
16 |
|
|
|
3 |
|
|
|
3 |
|
|
|
2 |
|
|||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||
i |
|
|
n |
|
79 |
|
|
|
|
79 |
|
|
79 |
|
|
|
|
79 |
|
|
|
|
|
79 |
|
|
|
79 |
|
|
79 |
|
|
79 |
|
||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||||||||||||||||||||
$4EE@4FD<64S QFBF EF4F<EF<K9E><= DS8 >4> ;4>BA |
|||||||||||||||||||||||||||||||||||||||||||||||||||||
D4ECD989?9A<S 8<E>D9FAB= E?GK4=AB= 69?<K<AO |
X , |
|
A4=89@ |
||||||||||||||||||||||||||||||||||||||||||||||||||
@4F9@4F<K9E>B9 B:<84A<9 M ( X ) < 8<EC9DE<R D( X ) QFB= |
|||||||||||||||||||||||||||||||||||||||||||||||||||||
E?GK4=AB= 69?<K<AO. !4IB8<@ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||||||||||||||||||||
M ( X ) ý 0 |
4 |
|
û1 |
13 |
û 2 |
14 |
û 3 |
24 |
û 4 |
16 |
|
û 5 |
3 |
û 7 |
2 |
ý |
|||||||||||||||||||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|||||||||||||||||||||||||||||||||||||||||||
|
|
|
|
|
79 |
|
|
|
|
79 |
|
79 |
|
|
|
79 |
|
|
|
79 |
|
|
79 |
|
|
|
|
|
79 |
|
|
|
|||||||||||||||||||||
ý 2.835 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||
!4=89@ 8<EC9DE<R CB HBD@G?9 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
m |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
|
|
|
|
|
|
|
|
D( X ) ý p(úi ) x(2i ) (M ( X )) 2 . |
|
|
|
|
|
|
|
|
|
|
||||||||||||||||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
i ý1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
#B QFB= HBD@G?9 A4IB8<@ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||||||||||||||||
D( X ) ý 0 2 |
4 |
|
û12 |
13 |
û 4 |
14 |
û 9 |
24 |
û16 |
16 |
û 25 |
3 |
û |
||||||||||||||||||||||||||||||||||||||||
|
|
|
|
|
|
|
|||||||||||||||||||||||||||||||||||||||||||||||
|
|
|
|
|
79 |
|
|
79 |
|
|
|
79 |
|
79 |
|
79 |
|
|
|
|
79 |
|
|
|
|
|
|
|
|||||||||||||||||||||||||
û 36 |
3 |
û 49 |
2 |
(2.835) 2 |
ý 2.37. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||||||||||||||||||||||||||
|
79 |
79 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||||
%?98B64F9?PAB, |
|
|
ED98A99 |
|
|
;A4K9A<9 < |
|
8<EC9DE<S, |
|||||||||||||||||||||||||||||||||||||||||||||
41
EBBF69FEF6GRM<9 D94?<;4J<< 6O5BD><, D46AO EBBF69FEF69AAB
x6 ý M ( X ) ý 2.835, d6 ý D( X ) ý 2.37.
42
& !"Ы% "
&"' '!/ "& !
#GEFP 6 ;4>BA D4ECD989?9A<S E?GK4=AB= 69?<K<AO X
6IB8<F C4D4@9FD ñ , ;A4K9A<9 >BFBDB7B A9<;69EFAB. @4F9@4F<K9E>B= EF4F<EF<>9 6 >4K9EF69 A9<;69EFAB7B
C4D4@9FD4 |
ñ |
<ECB?P;GRF K<E?B, |
CB?GK49@B9 A9>BFBDO@ |
|
B5D4;B@ (>4><@ <@9AAB – 5G89F D4EE@BFD9AB A<:9) CB |
||||
D94?<;4J<< 6O5BD>< |
(x1 , x2 ,...., xn ) , EBBF69FEF6GRM9= |
|||
E?GK4=AB= |
69?<K<A9 X . 4?99 D4EE@4FD<64RFES 6BCDBEO, |
|||
E6S;4AAO9 E A4IB:89A<9@ K<E?4, <ECB?P;G9@B7B 6 >4K9EF69 |
||||
;A4K9A<S A9<;69EFAB7B C4D4@9FD4 ñ . |
|
|||
"CD989?9A<9 |
|
|
|
|
#GEFP |
E?GK4=AB= |
69?<K<A9 X |
EBBF69FEF6G9F 6O5BD>4 |
|
( X1 , X 2 ,...., X n ) |
< 6 ;4>BA D4ECD989?9A<S E?GK4=AB= 69?<K<AO |
|||
X 6IB8<F A9<;69EFAO= C4D4@9FD ñ . &BK9KAB= BJ9A>B=
A9<;69EFAB7B C4D4@9FD4 ñ A4;O649FES E?GK4=A4S 69?<K<A4, S6?SRM4SES HGA>J<9= 6O5BD><
ñˆ ý ñ( X1 , X 2 ,..., X n ).
E?< QF4 HGA>J<S 6O5D4A4, FB 6 >4K9EF69 ;A4K9A<S A9<;69EFAB7B C4D4@9FD4 ñ CD<A<@49FES D94?<;4J<S QFB= E?GK4=AB= 69?<K<AO, EBBF69FEF6GRM4S D94?<;4J<< 6O5BD>< (x1 , x2 ,...., xn ) , FB 9EFP CB?474RF ñ ý ñ(x1 , x2 ,...., xn ) . BCDBE B
43
FB@ >4> 6O5<D4RF HGA>J<R ñ( X1 , X 2 ,...., X n ) 5G89F
D4EE@BFD9A A<:9.
4@9K4A<9
%GM9EF6G9F 59EK<E?9AAB9 @AB:9EF6B HGA>J<= 6O5BD><
ñˆ ý ñ( X1 , X 2 ,..., X n ).
9=EF6<F9?PAB, 6;S6 ?R5GR K<E?B6GR HGA>J<R n C9D9@9AAOI ñ(x1 , x2 ,...., xn ) < CB8EF46<6 6@9EFB K<E?B6OI
C9D9@9AAOI E?GK4=AO9 69?<K<AO ( X1 , X 2 ,...., X n ) , CB?GK<@
EBBF69FEF6GRMGR E?GK4=AGR 69?<K<AG
ñ( X1 , X 2 ,..., X n ).
45974S 6C9D98, E?98G9F BF@9F<FP, KFB >4><9-FB <; QF<I
HGA>J<= S6?SRFES 5B?99 CD98CBKF<F9?PAO@< E FBK>< ;D9A<S BJ9A>< A9<;69EFAB7B C4D4@9FD4 ñ . 484K< «D4A:<DB64A<S»
HGA>J<= |
ñˆ ý ñ( X1 , X 2 ,..., X n ) |
E FBK>< |
;D9A<S |
>4K9EF64 |
BJ9A>< C4D4@9FD4 ñ < A4IB:89A<S 5B?99 |
«>4K9EF69AAOI» |
|||
BJ9AB> 5G8GF D4EE@BFD9AO A<:9. |
|
|
|
|
4@9K4A<9 |
|
|
|
|
R5GR |
HGA>J<R 6O5BD>< |
( X1 , X 2 ,...., X n ) |
A4;O64RF |
|
EF4F<EF<>B=.
44
#D<@9D
#GEFP E?GK4=A4S 69?<K<A4 X <@99F ABD@4?PAB9 D4ECD989?9A<9 N (m;ó ) < A9<;69EFAO@ C4D4@9FDB@ S6?S9FES
@4F9@4F<K9E>B9 B:<84A<9 |
m . #GEFP E?GK4=AB= 69?<K<A9 X |
|||||
EBBF69FEF6G9F |
6O5BD>4 |
( X1 , X 2 ,...., X n ) |
< |
CB?GK9A4 |
||
D94?<;4J<S QFB= |
6O5BD>< |
(x1 , x2 ,...., xn ) . BCGEF<@, KFB 6 |
||||
>4K9EF69 BJ9A>< |
C4D4@9FD4 m 6O5D4A4 HGA>J<S |
6O5BD>< |
||||
6<84 |
|
|
|
|
|
|
|
|
|
1 |
n |
|
|
|
m( X n ) ý |
X k |
|
|
||
|
|
|
|
|||
|
|
|
n k ý1 |
|
|
|
(CBK9@G <@9AAB QF4 HGA>J<S – A9 B5EG:849FES). &B784 6 >4K9EF69 ;A4K9A<S A9<;69EFAB7B C4D4@9FD4 m 59D9FES K<E?B,
D46AB9
|
|
1 |
|
n |
|
|
m ý |
|
xk . |
|
|
|
|
|
|||
|
|
n k ý1 |
|
||
&4> >4> ñˆ ý ñ( X1 , X 2 ,..., X n ) 9EFP E?GK4=A4S 69?<K<A4, |
|||||
FB A9<;69EFAB@G C4D4@9FDG ñ |
EF46<FES 6 EBBF69FEF6<9 A9 |
||||
B8AB K<E?B, 4 @AB:9EF6B K<E9?, 4 <@9AAB, @AB:9EF6B |
|||||
D94?<;4J<= |
ñ(x1 , x2 ,...., xn ) |
E?GK4=AB= |
69?<K<AO |
||
ñˆ ý ñ( X1 , X 2 ,..., X n ) . ;46<E<@BEF< BF D94?<;4J<< 6O5BD><
(CB?GK9AAB= |
6 D9;G?PF4F9 Q>EC9D<@9AFB6) D94?<;4J<S |
ñ(x1 , x2 ,...., xn ) |
E?GK4=AB= 69?<K<AO ñˆ ý ñ( X1 , X 2 ,..., X n ) |
45
@B:9F ?<5B E<?PAB BF?<K4FPES BF <EF<AAB7B ;A4K9A<S ñ ,
?<5B A9 E<?PAB. BA9KAB, IBF9?BEP 5O, KFB5O D94?<;4J<S ñ(x1 , x2 ,...., xn ) >4> @B:AB K4M9 5O?4 5?<:9 > <EF<AAB@G
;A4K9A<R ñ |
< >4> @B:AB D9:9 ;A4K<F9?PAB BF?<K4?4EP BF |
|||
<EF<AAB7B ;A4K9A<S ñ . 'FB5O QFB <@9?B @9EFB, A9B5IB8<@B, |
||||
KFB5O E?GK4=A4S |
69?<K<A4 |
ñˆ ý ñ( X1 , X 2 ,..., X n ) |
CB |
|
6B;@B:ABEF< |
G8B6?9F6BDS?4 |
D4EE@4FD<649@O@ |
A<:9 |
|
GE?B6<S@. |
|
|
|
|
"CD989?9A<9 |
|
|
|
|
"J9A>4 |
ñˆ( X n ) |
A9<;69EFAB7B |
C4D4@9FD4 ñ A4;O649FES |
|
A9E@9M9AAB=, 9E?< 99 @4F9@4F<K9E>B9 B:<84A<9 EB6C4849F EB ;A4K9A<9@ QFB7B C4D4@9FD4
M (ñˆ( X n )) ý ñ.
?S FB7B, KFB5O 8B>4;4FP A9E@9M9AABEFP (<?< E@9M9AABEFP) BJ9A>< EB6E9@ A9 B5S;4F9?PAB ;A4FP ;A4K9A<9 ñ <
M (ñˆ( X n )) . !9E@9M9AABEFP BJ9A>< ñˆ( X n ) @B:AB 8B>4;4FP <ECB?P;GS E4@ 6<8 HGA>J<< ñˆ( X n ) . $4EE@BFD<@ QFB A4
CD<@9D9.
#D<@9D
#GEFP ( X1 , X 2 ,...., X n ) - 6O5BD>4, EBBF69FEF6GRM4S E?GK4=AB= 69?<K<A9 X . &B784 BJ9A>4
46
|
|
|
n |
||
|
|
( X n ) ý |
1 |
X k |
|
X |
|||||
|
|
||||
n k ý1
9EFP A9E@9M9AA4S BJ9A>4 @4F9@4F<K9E>B7B B:<84A<S E?GK4=AB= 69?<K<AO X , FB 9EFP
M ( 1 n X k ) ý M ( X ). n k ý1
E4@B@ 89?9, EB7?4EAB E6B=EF64@ @4F9@4F<K9E>B7B B:<84A<S
|
1 |
n |
1 |
n |
|
M ( |
X k ) ý |
M ( X k ) |
|||
|
|
||||
|
n k ý1 |
n k ý1 |
|||
89EP GKF9AB, KFB M ( X k ) ý M ( X ) .
"CD989?9A<9
"J9A>4 ñˆ( X n ) A9<;69EFAB7B
ý1 nM ( X ) ý M ( X ). n
C4D4@9FD4 ñ A4;O649FES
EBEFBSF9?PAB=, 9E?< BA4 CB 69DBSFABEF< EIB8<FES > A9<;69EFAB@G C4D4@9FDG ñ , FB 9EFP
lim P(ñˆ( X n ) ñ ü õ ) ý 1 CDи n .
%BEFBSF9?PABEFP BJ9A>< B;A4K49F, KFB CD< 8BEF4FBKAB 5B?PLB@ B5N9@9 6O5BD>< D94?<;4J<S BJ9A>< CBKF< 6E9784 5G89F E>B?P G7B8AB 5?<;>4 > C4D4@9FDG ñ (< FB?P>B <;D98>4 5G89F BF>?BASFPES A4 69?<K<AG 5B?PLGR õ ). &4> :9, >4> < 6
47
E?GK49 A9E@9M9AABEF<, 8?S CDB69D>< EBEFBSF9?PABEF< BJ9A>< ;A4A<S ;A4K9A<S A9<;69EFAB7B C4D4@9FD4 ñ A9 FD95G9FES.
"CD989?9A<9
!9E@9M9AA4S BJ9A>4 ñˆ( X n ) A4;O649FES QHH9>F<6AB=,
9E?< 99 8<EC9DE<S @<A<@4?PA4 CB ED46A9A<R E 8<EC9DE<S@< 8DG7<I A9E@9M9AAOI BJ9AB> CD< B8AB@ < FB@ :9 B5N9@9 6O5BD><.
$4A99 5O?B CB>4;4AB, KFB BJ9A>4
|
|
|
n |
||
|
|
( X n ) ý |
1 |
X k |
|
X |
|||||
|
|
||||
n k ý1
S6?S9FES A9E@9M9AAB= BJ9A>B= @4F9@4F<K9E>B7B B:<84A<S. B:AB CB>4;4FP (@O QFB7B 89?4FP A9 5G89@), KFB BJ9A>4
X ( X n ) S6?S9FES EBEFBSF9?PAB= BJ9A>B= @4F9@4F<K9E>B7B
B:<84A<S, 4 8?S ABD@4?PAB= E?GK4=AB= 69?<K<AO BA4 S6?S9FES < QHH9>F<6AB= BJ9A>B= @4F9@4F<K9E>B7B B:<84A<S.
B:AB CB>4;4FP, KFB 6O5BDBKA4S 8<EC9DE<S
|
|
n |
|||
D6 |
( X n ) ý |
1 |
( X k |
X |
) 2 , |
|
|||||
|
|
n k ý1 |
|||
789
48
X ý 1 n X k , n k ý1
S6?S9FES E@9M9AAB= BJ9A>B= 8<EC9DE<< E?GK4=AB= 69?<K<AO X , 4 <@9AAB, <@99F @9EFB E?98GRM99 EBBFABL9A<9
|
|
)) ý |
n 1 |
|
M (D ( X |
|
D( X ). |
||
n |
|
|||
6 |
|
n |
||
|
|
|
||
#BQFB@G 6 >4K9EF69 BJ9A>< 8<EC9DE<< E?GK4=AB= 69?<K<AO X <ECB?P;G9FES <ECD46?9AA4S 6O5BDBKA4S 8<EC9DE<S
|
|
) ý |
n |
|
|
|
|
S 2 ( X |
|
D ( X |
|
). |
|||
n |
|
|
n |
||||
|
|
n 1 |
6 |
|
|||
|
|
|
|
|
|
||
?S BJ9A>< S 2 ( X n ) <@99F @9EFB E?98GRM4S F9BD9@4.
&9BD9@4
ECD46?9AA4S 6O5BDBKA4S 8<EC9DE<S
|
|
) ý |
n |
|
|
|
|
S 2 ( X |
|
D ( X |
|
) |
|||
n |
|
|
n |
||||
|
|
n 1 |
6 |
|
|||
|
|
|
|
|
|
||
S6?S9FES A9E@9M9AAB= EBEFBSF9?PAB= BJ9A>B= 8<EC9DE<< E?GK4=AB= 69?<K<AO X .
!<:9 D4EE@4FD<64RFES @9FB8O A4IB:89A<S FBK9KAOI BJ9AB> (4 CB EGM9EF6G, @9FB8O A4IB:89A<S K<E?B6OI
49
HGA>J<= n C9D9@9AAOI), >BFBDO9 G8B6?9F6BDSRF G>4;4AAO@
6OL9 E6B=EF64@ (<?< IBFS 5O A9>BFBDO@ <; A<I).
#D9864D<F9?PAB ;4@9F<@ E?98GRM99. E?< <@99FES HGA>J<S 6O5BD>< ( X1 , X 2 ,..., X n ) , FB QFB= HGA>J<< @B:AB
CBEF46<FP 6 EBBF69FEF6<9 K<E?B6GR HGA>J<R (x1 , x2 ,..., xn ) K<E?B6OI C9D9@9AAOI (x1 , x2 ,...., xn ) . A4B5BDBF, 6ES>B= K<E?B6B= HGA>J<< n K<E?B6OI C9D9@9AAOI (x1 , x2 ,..., xn )
@B:AB CBEF46<FP 6 EBBF69FEF6<9 HGA>J<R 6O5BD><
( X1 , X 2 ,..., X n ) .
!4CD<@9D, CGEFP <@99FES HGA>J<S 6O5BD><
( X n ) ý X12 û X 22 û ... û X n2 .
E?< ;4@9A<FP E?GK4=AO9 69?<K<AO K<E?B6O@< C9D9@9AAO@< (x1 , x2 ,...., xn ) , FB CB?GK<@ K<E?B6GR HGA>J<R
(x1 , x2 ,..., xn ) ý x12 û x22 û ... û xn2
K<E?B6OI C9D9@9AAOI (x1 , x2 ,...., xn ) . A4B5BDBF, K<E?B6B= HGA>J<< n C9D9@9AAOI, A4CD<@9D, 6<84
(x1 , x2 ,..., xn ) ý x1 û x2 û... û xn
EBBF69FEF6G9F HGA>J<S 6O5BD>< 6<84
( X n ) ý X1 û X 2 û... û X n .
50
