Элементы математической статистики. Учебное пособие

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6O5BDBKAB= HGA>J<< ( X1 , X 2 ,..., X n ) < A4B5BDBF, C9D9IB8 BF

6O5BDBKAB= HGA>J<< > K<E?B6B=.

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$4EE@BFD<@ EA4K4?4 E?GK4=, >B784 X - A9CD9DO6A4S

E?GK4=A4S 69?<K<A4 < C?BFABEFP HGA>J<< D4ECD989?9A<S ;46<E<F BF B8AB7B A9<;69EFAB7B C4D4@9FD4 ñ , FB 9EFP <@99F 6<8 (ñ - A9<;69EFAO= C4D4@9FD). !4L4 J9?P –

CBEFDB<FP FBK9KAGR BJ9A>G QFB7B A9<;69EFAB7B C4D4@9FD4. &BK9KA4S BJ9A>4 ñˆ( X n ) A9<;69EFAB7B C4D4@9FD4 ñ CB @9FB8G

@B@9AFB6 EFDB<FES E?98GRM<@ B5D4;B@.

4> <;69EFAB, @4F9@4F<K9E>B9 B:<84A<9 A9CD9DO6AB= E?GK4=AB= 69?<K<AO X D46AB

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ñ -A9<;69EFAO= C4D4@9FD. #B>4:9@, KFB @4F9@4F<K9E>B9

#D989? ;89EP 59D9FES CD< û. &4><@ B5D4;B@, 6 84AAB@

E?GK49 CB?GK49@

M ( X ) ý (ñ ) ý ñ1 .

&9C9DP CDB8B?:<@ CBEFDB9A<9 BJ9A>< A9<;69EFAB7B C4D4@9FD4 @9FB8B@ @B@9AFB6. #GEFP E?GK4=AB= 69?<K<A9 X EBBF69FEF6G9F 6O5BD>4 ( X1 , X 2 ,...., X n ) . $4A99 5O?4

BCD989?9A4 E?GK4=A4S 69?<K<A4 X ( X n ) - 6O5BDBKAB9 ED98A99

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1FB= HGA>J<< 6O5BD>< @B:AB CBEF46<FP 6 EBBF69FEF6<9 K<E?B6GR HGA>J<R n C9D9@9AAOI

( x1 , x2 ,..., xn ) ý 1 ( x1 û x2 û ... û xn ), n

>BFBD4S CB?GK49FES ;4@9AB= 6 6O5BDBKAB= HGA>J<< E?GK4=AOI 69?<K<A X 1 , X 2 ,...., X n K<E?B6O@< C9D9@9AAO@<

x1 , x2 ,...., xn . #D<D46AS9@ HBD@G?G, BCD989?SRMGR HGA>J<R

(x1 , x2 ,..., xn ) HBD@G?9, BCD989?SRM9= HGA>J<R (ñ ) ;

CB?GK<@ D469AEF6B

(ñ ) ý 1 ( x1 û x2 û ... û xn ). n

?<

(ñ) 1 ( x1 û x2 û ... û xn ) ý 0, n

>BFBDB9 @B:AB D4EE@4FD<64FP >4> GD46A9A<9 E (n û1) A9<;69EFAO@< (x1 , x2 ,..., xn ;ñ)

F (x1 , x2 ,..., xn ;ñ) ý 0.

1FB GD46A9A<9 BCD989?S9F A9<;69EFAB9 ñ >4> A9S6AGR HGA>J<R C9D9@9AAOI (x1 , x2 ,...., xn ) . "5B;A4K<@ QFG A9S6AGR

54

HGA>J<R K<E?B6OI C9D9@9AAOI E<@6B?B@ ñ(x1 , x2 ,...., xn ) .

B:AB ;4C<E4FP

ñ ý ñ(x1 , x2 ,...., xn ).

E?< F9C9DP K<E?B6O9 C9D9@9AAO9 (x1 , x2 ,...., xn ) ;4@9A<FP 6O5BD>B= ( X1 , X 2 ,...., X n ) , FB CB?GK<@ E?GK4=AGR 69?<K<AG

ñˆ ý ñ( X1 , X 2 ,...., X n ),

S6?SRMGRES HGA>J<9= 6O5BD><. 1F4 E?GK4=A4S 69?<K<A4 A4;O649FES FBK9KAB= BJ9A>B= C4D4@9FD4 ñ , CB?GK9AAB= @9FB8B@ @B@9AFB6.

$4EE@BFD<@ CD<@9D A4 CBEFDB9A<9 FBK9KAB= BJ9A>< CB @9FB8G @B@9AFB6.

#D<@9D

%?GK4=A4S 69?<K<A4 X D4ECD989?9A4 CB CB>4;4F9?PAB@G

789 ü þ 0 - A9<;69EFAO= C4D4@9FD (C4D4@9FD CB>4;4F9?PAB7B

D4ECD989?9A<S FD48<J<BAAB B5B;A4K49FES 5G>6B= ü ). !4=F< FBK9KAGR BJ9A>G A9<;69EFAB7B C4D4@9FD4 ü CB @9FB8G @B@9AFB6.

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$9L9A<9

$4A99 6 CD<@9D9 A4 EFD4A<J9 51 5O?B CB?GK9AB 6OD4:9A<9 8?S @4F9@4F<K9E>B7B B:<84A<S M ( X ) >4>

HGA>J<< A9<;69EFAB7B C4D4@9FD4

M ( X ) ý (ü) ý ü1 ,

(6 CD<@9D9 C4D4@9FD B5B;A4K4?ES 5G>6B= ñ ). #D<D46A<64S HGA>J<< (ü) <

( x1 , x2 ,..., xn ) ý 1 ( x1 û x2 û ... û xn ), n

CB?GK49@ GD46A9A<9

56

(L?SC>4 A48 5G>6B= ü CBEF46?9A4 8?S FB7B, KFB5O

CB8K9D>AGFP, KFB ;89EP üˆ S6?S9FES E?GK4=AB= 69?<K<AB=), 789

Xý 1 ( X1 û X 2 û ... û X n ) n

9EFP 6O5BDBKAB9 ED98A99. E?< x6 9EFP D94?<;4J<S

6O5BDBKAB7B ED98A97B, EBBF69FEF6GRM4S D94?<;4J<< 6O5BD>< (x1 , x2 ,...., xn ) , FB D94?<;4J<S E?GK4=AB= 69?<K<AO üˆ ,

EBBF69FEF6GRM4S D94?<;4J<< 6O5BD>< (>BFBDGR B5B;A4K<@ ü6 ), D46A4

ü6 ý 1 . x6

&9C9DP D4EE@BFD<@ CD<@9D A4IB:89A<S CD<5?<:9AAB7B ;A4K9A<S A9<;69EFAB7B C4D4@9FD4 ü CB>4;4F9?PAB7B D4ECD989?9A<S CB @9FB8G @B@9AFB6.

#D<@9D

%?GK4=A4S 69?<K<A4 X D4ECD989?9A4 CB CB>4;4F9?PAB@G

;4>BAG E C?BFABEFPR

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789 ü þ 0 - A9<;69EFAO= C4D4@9FD. #B?GK9A4 6O5BD>4 B5N9@4 n ý 200. #B 6O5BD>9 CBEFDB9A EF4F<EF<K9E><= DS8 K4EFBF

û 22.5 2 û 27.5 1) ý 5

&B784 ü6 , >BFBDB9 < CD<A<@49FES ;4 CD<5?<:9AAB9 ;A4K9A<9 ü , 5G89F D46AB (E@BFD< CD98O8GM<= CD<@9D)

ü6 ý 1 ý 1 ý 0.2. x6 5

B5M9@ E?GK49 C?BFABEFP HGA>J<< D4ECD989?9A<S E?GK4=AB= 69?<K<AO X @B:9F ;46<E9FP A9 BF B8AB7B, 4 BF

A9E>B?P><I A9<;69EFAOI C4D4@9FDB6. #GEFP, A4CD<@9D,

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C?BFABEFP p(x;ñ1 ,ñ2 ) ;46<E<F BF 86GI A9<;69EFAOI C4D4@9FDB6 ñ1 < ñ2 . QFB@ E?GK49 A4IB8<@ 6OD4:9A<S 8?S

@4F9@4F<K9E>B7B B:<84A<S < 8<EC9DE<<. 6OD4:9A<S 8?S M ( X ) < D( X ) 5G8GF 6IB8<FP C4D4@9FDO ñ1 < ñ2 .

û

M ( X ) ý xp( x;ñ1 ,ñ2 )dx ý 1 (ñ1 ,ñ2 ),

û

D( X ) ý x2 p( x;ñ1 ,ñ2 )dx ( 1 (ñ1 ,ñ2 ))2 ý 2 (ñ1 ,ñ2 ) .

O5BDBKAB@G ED98A9@G

X ý 1 n X k n k ý1

< 6O5BDBKAB= 8<EC9DE<<

EBBF69FEF6GRF K<E?B6O9 HGA>J<<

1 ( x1 , x2 ,..., xn ) ý 1 n xk n k ý1

<

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HGA>J<< BF (x1 , x2 ,...., xn ) . %?98B64F9?PAB, <@99@ 869 A9S6AB

;484AAO9 K<E?B6O9 HGA>J<<

6O5BDBKAO9 ( X1 , X 2 ,...., X n ) , FB CB?GK<@ 869 E?GK4=AO9

69?<K<AO

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