Problem

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

$$G_X(t) = k(1 + 3t + 4t^2),$$

where $k$ is a constant.

(a) Show that $E(X) = \dfrac{11}{8}$.

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

$$G_Y(t) = \frac{1}{3}t^2(1 + 2t).$$

The random variables $X$ and $Y$ are independent and $Z = X + Y$.

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

(c) Use your answer to part (b) to find the value of $\operatorname{Var}(Z)$.

(d) Write down the most probable value of $Z$.