# Question: Use Theorem 4 14 on page 135 and its corollary to

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 < j.

(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 < j.

## Relevant Questions

Prove Theorem 8.9. Theorem 8.9 If X1, X2, . . . , Xn are independent random variables having chi- square distributions with v1, v2, . . . , vn degrees of freedom, then Has the chi-square distribution with v1 + v2 + · · · ...Based on the result of Exercise 8.24, show that if X is a random variable having a chi– square distribution with v degrees of freedom and v is large, the distribution of X – v / √2v can be approximated with the ...Use the transformation technique based on Theorem 7.2 on page 218 to rework the proof of Theorem 8.14. Let f = u/v1 v/v2 and w = v.) Show that the F distribution with 4 and 4 degrees of freedom is given by And use this density to find the probability that for independent random samples of size n = 5 from normal populations with the same variance, S21 / ...Duplicate the method used in the proof of Theorem 8.16 to show that the joint density of Y1 and Yn is given by (a) Use this result to find the joint density of Y1 and Yn for random samples of size n from an exponential ...Post your question