Problem

A 6-sided dice, $A$, with faces numbered 1, 2, 3, 4, 5, 6 is biased so that the probability of throwing a 6 is $\frac{1}{4}$. The random variable $X$ is the number of 6s obtained when dice $A$ is thrown twice.

(a) Find the probability generating function of $X$.

A second dice, $B$, with faces numbered 1, 2, 3, 4, 5, 6 is unbiased. The random variable $Y$ is the number of 6s obtained when dice $B$ is thrown twice.

The random variable $Z$ is the total number of 6s obtained when both dice are thrown twice.

(b) Find the probability generating function of $Z$, expressing your answer as a polynomial.

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

(d) Use the probability generating function of $Z$ to find the most probable value of $Z$.