Page 351 - DMTH404_STATISTICS

Unit 24: Sampling Distributions

In the case of sampling without replacement, we can write Notes

N N

P -
P -
Cov(X , X) = E(X - m)(X - m) = å å ( r m )( s m ) p× rs ,
i j i j
=
=
r 1 s 1,

s r
where p is the joint probability that the rth unit of population is drawn at the ith draw and the
rs
1
sth unit of population is drawn at the jth draw. We note that p = . Thus, we have
rs
N (N 1- )
N N 1
P -
Cov ( X X, j ) = å å ( r m )( s ) m ×

P -
i
=
=
r 1 s 1, N (N 1- )
s r

1 N N
= å ( r m ) å ( s ) m
P -
P -
N (N 1- ) r 1= s 1,
=
s r

1 N é N ù
P -
= å ( r m ) å ( s m - P - m ) ú
P -
ê
) ( r
N (N 1- ) r 1= ë s 1 û
=
1 N
P -
= å ( r m ) 0 é - ( r ) m ù
P -
N (N 1- ) r 1= ë û
1 N 2 1 2 s 2
= - å ( r ) m = - × Ns = -
P -
N (N 1- ) r 1= N (N 1- ) (N 1- )
24.2 Sampling Distribution of Sample Mean
X + X +  + X n
1
2
We know that X = . In the previous section we have shown that if the
n
sample is random, then each of the X 's are random variable with mean m and variance s . Since
2
i
X is a linear combination of these random variables, therefore, it is also a random variable with
1 1
mean equal to ( ) = é ë E X 1 ( ) +  + E X n û × nm = m and variance equal to
( ) ù =
E X
( ) E X+
2
n n
2
2 é X + X +  + X n ù
2
1
( ) ( X m=
Var X E - ) = E ê - m ú
ë n û
2
é (X + X +  + X n ) nmù 1 2
-
2
1
= E ê ú = 2 E éå (X - ) m ù û
i
ë
ê ë n ú û n
1 é 2 ù
= 2 E êå (X - ) m + å å (X - m )( X - ) m ú
i
i
j
n ê ë i j ú û

1 é 2 ù
= êå E (X - ) m + å å E (X - m )( X - ) m ú
2 i i j
n ê ë i j ú û

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