Problem

The random variable $Y$ is the sum of two independent observations of the random variable $X$. The probability generating function $G_Y(t)$ of $Y$ is given by

$$G_Y(t) = \frac{t^2}{(4 - 3t)^4}.$$

(a) Find $E(Y)$.

(b) Write down an expression for the probability generating function of $X$.

(c) Find $P(X = 4)$.