Problem

The discrete random variable $X$ has probability generating function $G_X(t)$ given by

$$G_X(t)=\frac{t}{(3-2t)^2}.$$

(a) Find $\mathrm{E}(X)$ and $\operatorname{Var}(X)$.

The discrete random variable $Y$ has probability generating function $G_Y(t)$ given by

$$G_Y(t)=\frac{t^2}{(3-2t)^2}.$$

The random variable $Z$ is the sum of the random variables $X$ and $Y$.

(b) Assuming $X$ and $Y$ are independent, find $P(Z > 4)$.