Problem

Jason has three biased coins. For each coin the probability of obtaining a head when it is thrown is $\frac{2}{3}$. Jason throws all three coins. The number of heads obtained is denoted by $X$.

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

Jason also has two unbiased coins. He throws all five coins. The number of heads obtained from the two unbiased coins is denoted by $Y$. It is given that $G_Y(t)=\frac{1}{4}+\frac{1}{2}t+\frac{1}{4}t^2$. The random variable $Z$ is the total number of heads obtained when Jason throws all five coins.

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

(c) Find $\operatorname{E}(Z)$.