5 The discrete random variable \(U\) has probability distribution given by
$$\mathrm { P } ( U = r ) = \begin{cases} \frac { 1 } { 16 } \binom { 4 } { r } & r = 0,1,2,3,4
0 & \text { otherwise } \end{cases}$$
- Find and simplify the probability generating function (pgf) of \(U\).
- Use the pgf to find \(\mathrm { E } ( U )\) and \(\operatorname { Var } ( U )\).
- Identify the distribution of \(U\), giving the values of any parameters.
- Obtain the pgf of \(Y\), where \(Y = U ^ { 2 }\).
- State, giving a reason, whether you can obtain the pgf of \(U + Y\) by multiplying the pgf of \(U\) by the pgf of \(Y\).