4 The random variable \(X\) has probability generating function \(\mathrm { G } _ { X } ( t )\) given by
$$\mathrm { G } _ { X } ( t ) = \operatorname { ct } ( 1 + t ) ^ { 5 }$$
where \(c\) is a constant.
- Find the value of \(c\).
- Find the value of \(\mathrm { E } ( X )\).
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The random variable \(Y\) is the sum of two independent values of \(X\). - Write down the probability generating function of \(Y\) and hence find \(\operatorname { Var } ( Y )\).
- Find \(\mathrm { P } ( Y = 5 )\).