Let [math]X[/math] and [math]Y[/math] be random variables with values in [math]\{1,2,3,4,5,6\}[/math] with distribution functions [math]p_X[/math] and [math]p_Y[/math] given by

[[math]] \begin{eqnarray*} p_X(j) &=& a_j\ , \\ p_Y(j) &=& b_j\ . \end{eqnarray*} [[/math]]

• Find the ordinary generating functions [math]h_X(z)[/math] and [math]h_Y(z)[/math] for these distributions.
• Find the ordinary generating function [math]h_Z(z)[/math] for the distribution [math]Z = X + Y[/math].
• Show that [math]h_Z(z)[/math] cannot ever have the form
  [[math]] h_Z(z) = \frac{z^2 + z^3 +\cdots+ z^{12}}{11}\ . [[/math]]

Hint: [math]h_X[/math] and [math]h_Y[/math] must have at least one nonzero root, but [math]h_Z(z)[/math] 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).