2 The random variable \(X\) has the \(\chi _ { n } ^ { 2 }\) distribution. This distribution has moment generating function \(\mathrm { M } ( \theta ) = ( 1 - 2 \theta ) ^ { - \frac { 1 } { 2 } n }\), where \(\theta < \frac { 1 } { 2 }\).

1. Verify the expression for \(\mathrm { M } ( \theta )\) quoted above for the cases \(n = 2\) and \(n = 4\), given that the probability density functions of \(X\) in these cases are as follows. $$\begin{array} { l l } n = 2 : & \mathrm { f } ( x ) = \frac { 1 } { 2 } \mathrm { e } ^ { - \frac { 1 } { 2 } x } \quad ( x > 0 )
   n = 4 : & \mathrm { f } ( x ) = \frac { 1 } { 4 } x \mathrm { e } ^ { - \frac { 1 } { 2 } x } \quad ( x > 0 ) \end{array}$$
2. For the general case, use \(\mathrm { M } ( \theta )\) to find the mean and variance of \(X\) in terms of \(n\).
3. \(Y _ { 1 } , Y _ { 2 } , \ldots , Y _ { k }\) are independent random variables, each with the \(\chi _ { 1 } ^ { 2 }\) distribution. Show that \(W = \sum _ { i = 1 } ^ { k } Y _ { i }\) has the \(\chi _ { k } ^ { 2 }\) distribution.
4. Use the Central Limit Theorem to find an approximation for \(\mathrm { P } ( W < 118.5 )\) for the case \(k = 100\).