Problem

$Y$ is a discrete random variable which takes the values $0, 2, 4, \ldots$. The probability generating function of $Y$ is given by

$$G_Y(t) = \frac{k}{1 - at^2}.$$

(a) Find $k$ in terms of $a$.

(b) Show that $P(Y > 2) = a^2$.

It is now given that $a = 0.2$.

(c) Find the value of $E(Y)$.