Problem

The random variable $X$ takes values 1 and 2 with probabilities $\frac{2}{5}$ and $\frac{3}{5}$ respectively.

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

The random variable $Y$ is the sum of four independent observations of $X$.

(b) Find the probability generating function $G_Y(t)$ of $Y$. Give your answer in the form $G_Y(t)=at^m(b+ct)^n$, where $a$, $b$, $c$, $m$ and $n$ are constants to be determined.

(c) Use $G_Y(t)$ to find $\mathrm{P}(Y=6)$.

(d) Find $\operatorname{Var}(Y)$.