Problem

Eric has three identical coins, each of which is biased so that the probability of obtaining a head when it is thrown is $\frac{1}{3}$. The random variable $X$ is the number of heads obtained when Eric throws the three coins at the same time.

(a) Find the probability generating function $G_X(t)$ of $X$.

Eric also has two fair 6-sided dice with faces numbered 1 to 6. The random variable $Y$ is the number of sixes obtained when Eric throws the two dice at the same time. It is given that the probability generating function of $Y$ is $\frac{25}{36}+\frac{10}{36}t+\frac{1}{36}t^2$.

Eric throws the three coins and the two dice. The random variable $Z$ is the sum of the number of heads obtained and the number of sixes obtained.

(b) Find the probability generating function $G_Z(t)$ of $Z$, expressing your answer as a polynomial in $t$.

(c) Use $G_Z(t)$ to find $E(Z)$ and $\operatorname{Var}(Z)$.