Let X and Y be random variables with values in {1, 2, 3, 4, 5, 6} with distribution functions PX and PY given by
PX (j) = a j, PY (j) = b j .
(a) Find the ordinary generating functions hX (z) and hY (z) for these distributions.
(b) Find the ordinary generating function hZ(z) for the distribution Z = X + Y .
(c) Show that hZ (z) cannot ever have the form

Hint: hX and hY must have at least one nonzero root, but hZ (z) in the form given has no nonzero real roots. It follows from this observation that there is no way to load two dice so that the probability that a given sum will turn up when they are tossed is the same for all sums (i.e., that all outcomes are equally likely).

22 + 23 + ...+z12 hz(2): 11 11