Problem

George throws two coins, A and B, at the same time. Coin A is biased so that the probability of obtaining a head is $a$. Coin B is biased so that the probability of obtaining a head is $b$, where $b < a$. The probability generating function of $X$, the number of heads obtained by George, is $G_X(t)$. The coefficients of $t$ and $t^2$ in $G_X(t)$ are $\frac{5}{12}$ and $\frac{1}{12}$ respectively.

(a) Find the value of $a$.

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

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

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