Problem

Harry has three coins.

• One coin is biased so that, when it is thrown, the probability of obtaining a head is $\frac{1}{3}$.
• The second coin is biased so that, when it is thrown, the probability of obtaining a head is $\frac{1}{4}$.
• The third coin is biased so that, when it is thrown, the probability of obtaining a head is $\frac{1}{5}$.

The random variable $X$ is the number of heads that Harry obtains when he throws all three coins together.

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

Isaac has two fair coins. The random variable $Y$ is the number of heads that Isaac obtains when he throws both of his coins together. The random variable $Z$ is the total number of heads obtained when Harry throws his three coins and Isaac throws his two coins.

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

(c) Use the probability generating function of $Z$ to find $\mathrm{E}(Z)$.