2 The random variable \(Z\) has the standard Normal distribution. The random variable \(Y\) is defined by \(Y = Z ^ { 2 }\).
You are given that \(Y\) has the following probability density function. $$\mathrm { f } ( y ) = \frac { 1 } { \sqrt { 2 \pi y } } \mathrm { e } ^ { - \frac { 1 } { 2 } y } , \quad y > 0$$

1. Show that the moment generating function (mgf) of \(Y\) is given by $$\mathrm { M } _ { Y } ( \theta ) = ( 1 - 2 \theta ) ^ { - \frac { 1 } { 2 } }$$
2. Use the mgf to obtain \(\mathrm { E } ( Y )\) and \(\operatorname { Var } ( Y )\). The random variable \(U\) is defined by $$U = Z _ { 1 } ^ { 2 } + Z _ { 2 } ^ { 2 } + \ldots + Z _ { n } ^ { 2 } ,$$ where \(Z _ { 1 } , Z _ { 2 } , \ldots , Z _ { n }\) are independent standard Normal random variables.
3. State an appropriate general theorem for mgfs and hence write down the mgf of \(U\). State the values of \(\mathrm { E } ( U )\) and \(\operatorname { Var } ( U )\). The random variable \(W\) is defined by $$W = \frac { U - n } { \sqrt { 2 n } }$$
4. Show that the logarithm of the \(\operatorname { mgf }\) of \(W\) is $$- \sqrt { \frac { n } { 2 } } \theta - \frac { n } { 2 } \ln \left( 1 - \sqrt { \frac { 2 } { n } } \theta \right) .$$ Use the series expansion of \(\ln ( 1 - t )\) to show that, as \(n \rightarrow \infty\), this expression tends to \(\frac { 1 } { 2 } \theta ^ { 2 }\).
   State what this implies about the distribution of \(W\) for large \(n\).