5 The continuous random variable \(X\) has probability density function given by
$$\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { ( \alpha - 1 ) ! } x ^ { \alpha - 1 } \mathrm { e } ^ { - x } & x \geqslant 0 \\ 0 & x < 0 \end{cases}$$
where \(\alpha\) is a positive integer.
- Explain how you can deduce that \(\int _ { 0 } ^ { \infty } x ^ { \alpha - 1 } \mathrm { e } ^ { - x } \mathrm {~d} x = ( \alpha - 1 )\) !.
- Write down an integral for the moment generating function \(\mathrm { M } _ { X } ( t )\) of \(X\) and show, by using the substitution \(x = \frac { u } { 1 - t }\), that \(\mathrm { M } _ { X } ( t ) = ( 1 - t ) ^ { - \alpha }\).
- Use the moment generating function to find, in terms of \(\alpha\),
- \(\mathrm { E } ( X )\),
- \(\operatorname { Var } ( X )\).