(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 1 and 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 2 Independence 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
But and are not independent, as for example