Problem

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

$$G_X(t) = ct(1 + t)^5,$$

where $c$ is a constant.

(a) Find the value of $c$.

(b) Find the value of $E(X)$.

The random variable $Y$ is the sum of two independent values of $X$.

(c) Write down the probability generating function of $Y$ and hence find $\operatorname{Var}(Y)$.

(d) Find $P(Y = 5)$.