Use Theorem 4.14 on page 135 and its corollary to show that if X11, X12, . . . , X1n1 , X21, X22, . . . , X2n2 are independent random variables, with the first n1 constituting a random sample from an infinite population with the mean µ1 and the variance σ21 and the other n2 constituting a random sample from an infinite population with the mean µ2 and the variance σ22 , then
(a) E(1 €“ 2) = µ1 €“ µ2;
(b) var(1 €“ 2) = σ21/n1 + σ22/n2 .
Theorem 4.14
If X1, X2, . . . , Xn are random variables and

Where a1, a2, . . . , an are constants, then

And

Where the double summation extends over all values of i and j, from 1 to n, for which i

Transcribed Image Text:

12 i-1 rt var( Y) = Σ a, . var(Xi) + 2 i-1 sy