2 The random variable \(X\) has the binomial distribution with parameters \(n\) and \(p\), i.e. \(X \sim \mathrm {~B} ( n , p )\).
- Show that the probability generating function of \(X\) is \(\mathrm { G } ( t ) = ( q + p t ) ^ { n }\), where \(q = 1 - p\).
- 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 \(X\) and use the linear transformation result to show that the moment generating function of \(Z\) is
$$\mathrm { M } _ { Z } ( \theta ) = \left( q \mathrm { e } ^ { - \frac { p \theta } { \sqrt { n p q } } } + p \mathrm { e } ^ { \frac { q \theta } { \sqrt { n p q } } } \right) ^ { n } .$$
- By expanding the exponential terms in \(\mathrm { M } _ { Z } ( \theta )\), show that the limit of \(\mathrm { M } _ { Z } ( \theta )\) as \(n \rightarrow \infty\) is \(\mathrm { e } ^ { \theta ^ { 2 } / 2 }\). You may use the result \(\lim _ { n \rightarrow \infty } \left( 1 + \frac { y + \mathrm { f } ( n ) } { n } \right) ^ { n } = \mathrm { e } ^ { y }\) provided \(\mathrm { f } ( n ) \rightarrow 0\) as \(n \rightarrow \infty\).
- What does the result in part (v) imply about the distribution of \(Z\) as \(n \rightarrow \infty\) ? Explain your reasoning briefly.
- What does the result in part (vi) imply about the distribution of \(X\) as \(n \rightarrow \infty\) ?