2 The random variable \(X\) takes values \(- 2,0\) and 2 , each with probability \(\frac { 1 } { 3 }\).
- Write down the values of
(A) \(\mu\), the mean of \(X\),
(B) \(\mathrm { E } \left( X ^ { 2 } \right)\),
(C) \(\sigma ^ { 2 }\), the variance of \(X\). - Obtain the moment generating function (mgf) of \(X\).
A random sample of \(n\) independent observations on \(X\) has sample mean \(\bar { X }\), and the standardised mean is denoted by \(Z\) where
$$Z = \frac { \bar { X } - \mu } { \frac { \sigma } { \sqrt { n } } }$$
- Stating carefully the required general results for mgfs of sums and of linear transformations, show that the mgf of \(Z\) is
$$M _ { Z } ( \theta ) = \left\{ \frac { 1 } { 3 } \left( 1 + e ^ { \frac { \theta \sqrt { 3 } } { \sqrt { 2 n } } } + e ^ { - \frac { \theta \sqrt { 3 } } { \sqrt { 2 n } } } \right) \right\} ^ { n } .$$
- By expanding the exponential functions in \(\mathrm { M } _ { Z } ( \theta )\), show that, for large \(n\),
$$\mathrm { M } _ { Z } ( \theta ) \approx \left( 1 + \frac { \theta ^ { 2 } } { 2 n } \right) ^ { n }$$
- Use the result \(\mathrm { e } ^ { y } = \lim _ { n \rightarrow \infty } \left( 1 + \frac { y } { n } \right) ^ { n }\) to find the limit of \(\mathrm { M } _ { Z } ( \theta )\) as \(n \rightarrow \infty\), and deduce the approximate distribution of \(Z\) for large \(n\).