7 The probability generating function of the random variable \(X\) is given by
$$\mathrm { G } ( t ) = \frac { 1 + a t } { 4 - t }$$
where \(a\) is a constant.
- Find the value of \(a\).
- Find \(\mathrm { P } ( X = 3 )\).
The sum of 3 independent observations of \(X\) is denoted by \(Y\). The probability generating function of \(Y\) is denoted by \(\mathrm { H } ( t )\).
- Use \(\mathrm { H } ( t )\) to find \(\mathrm { E } ( Y )\).
- By considering \(\mathrm { H } ( - 1 ) + \mathrm { H } ( 1 )\), show that \(\mathrm { P } ( Y\) is an even number \() = \frac { 62 } { 125 }\).
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