Problem

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

$$G_X(t) = \frac{1}{5} + pt + qt^2,$$

where $p$ and $q$ are constants.

(a) Given that $\mathrm{E}(X) = 1.1$, find the numerical value of $\mathrm{Var}(X)$.

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

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

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

(b) Find the probability generating function of $Z$.

(c) Find $\mathrm{P}(Z > 2)$.

(d) State the most probable value of $Z$.