2 [In this question, you may use the result \(\int _ { 0 } ^ { \infty } u ^ { m } \mathrm { e } ^ { - u } \mathrm {~d} u = m\) ! for any non-negative integer \(m\).]
The random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \begin{cases} \frac { \lambda ^ { k + 1 } x ^ { k } \mathrm { e } ^ { - \lambda x } } { k ! } , & x > 0
0 , & \text { elsewhere } \end{cases}$$ where \(\lambda > 0\) and \(k\) is a non-negative integer.

1. Show that the moment generating function of \(X\) is \(\left( \frac { \lambda } { \lambda - \theta } \right) ^ { k + 1 }\).
2. The random variable \(Y\) is the sum of \(n\) independent random variables each distributed as \(X\). Find the moment generating function of \(Y\) and hence obtain the mean and variance of \(Y\). [8]
3. State the probability density function of \(Y\).
4. For the case \(\lambda = 1 , k = 2\) and \(n = 5\), it may be shown that the definite integral of the probability density function of \(Y\) between limits 10 and \(\infty\) is 0.9165 . Calculate the corresponding probability that would be given by a Normal approximation and comment briefly.