Problem

The probability generating function, $G_Y(t)$, of the random variable $Y$ is given by

$G_Y(t) = 0.04 + 0.2t + 0.37t^2 + 0.3t^3 + 0.09t^4.$

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

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

(b) Find the probability generating function of $X$, giving your answer as a polynomial in $t$.