4 Jason has three biased coins. For each coin the probability of obtaining a head when it is thrown is \(\frac { 2 } { 3 }\). Jason throws all three coins. The number of heads obtained is denoted by \(X\).

1. Find the probability generating function \(\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )\) of \(X\).
   Jason also has two unbiased coins. He throws all five coins. The number of heads obtained from the two unbiased coins is denoted by \(Y\). It is given that \(G _ { Y } ( t ) = \frac { 1 } { 4 } + \frac { 1 } { 2 } t + \frac { 1 } { 4 } t ^ { 2 }\). The random variable \(Z\) is the total number of heads obtained when Jason throws all five coins.
2. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
3. Find \(\mathrm { E } ( \mathrm { Z } )\).