(This problem is from *A First Course in Stochastic Processes* by Karlin and Taylor.) Let and be jointly distributed nonnegative integer-valued random variables. For , define the joint generating function

and the marginal generating functions

Theorem 1and are independent if and only if the joint generating function is equal to the product of the marginal generating functions.

*Proof:* For the “only if” implication, we have

For the “if” implication, from

for all and we see that for all and , and hence and are independent.

Theorem 2Independence is sufficient, but not necessary for the generating function of to be the product of the marginal generating functions of and .

*Proof:* Let and each be uniformly distributed over , with joint distribution

Then

and

But and are not independent, as for example