2 The random variable \(X\) has the Poisson distribution with parameter \(\lambda\).
- Show that the probability generating function of \(X\) is \(\mathrm { G } ( t ) = \mathrm { e } ^ { \lambda ( t - 1 ) }\).
- Hence obtain the mean \(\mu\) and variance \(\sigma ^ { 2 }\) of \(X\).
- Write down the mean and variance of the random variable \(Z = \frac { X - \mu } { \sigma }\).
- Write down the moment generating function of \(X\). State the linear transformation result for moment generating functions and use it to show that the moment generating function of \(Z\) is
$$\mathrm { M } _ { Z } ( \theta ) = \mathrm { e } ^ { \mathrm { f } ( \theta ) } \quad \text { where } \mathrm { f } ( \theta ) = \lambda \left( \mathrm { e } ^ { \theta / \sqrt { \lambda } } - \frac { \theta } { \sqrt { \lambda } } - 1 \right)$$
- Show that the limit of \(\mathrm { M } _ { Z } ( \theta )\) as \(\lambda \rightarrow \infty\) is \(\mathrm { e } ^ { \theta ^ { 2 } / 2 }\).
- Explain briefly why this implies that the distribution of \(Z\) tends to \(\mathrm { N } ( 0,1 )\) as \(\lambda \rightarrow \infty\). What does this imply about the distribution of \(X\) as \(\lambda \rightarrow \infty\) ?