2 The random variable $X$ takes values $- 2,0$ and 2 , each with probability $\frac { 1 } { 3 }$.
\begin{enumerate}[label=(\roman*)]
\item Write down the values of\\
(A) $\mu$, the mean of $X$,\\
(B) $\mathrm { E } \left( X ^ { 2 } \right)$,\\
(C) $\sigma ^ { 2 }$, the variance of $X$.
\item 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

$$Z = \frac { \bar { X } - \mu } { \frac { \sigma } { \sqrt { n } } }$$
\item Stating carefully the required general results for mgfs of sums and of linear transformations, show that the mgf of $Z$ is

$$M _ { Z } ( \theta ) = \left\{ \frac { 1 } { 3 } \left( 1 + e ^ { \frac { \theta \sqrt { 3 } } { \sqrt { 2 n } } } + e ^ { - \frac { \theta \sqrt { 3 } } { \sqrt { 2 n } } } \right) \right\} ^ { n } .$$
\item By expanding the exponential functions in $\mathrm { M } _ { Z } ( \theta )$, show that, for large $n$,

$$\mathrm { M } _ { Z } ( \theta ) \approx \left( 1 + \frac { \theta ^ { 2 } } { 2 n } \right) ^ { n }$$
\item 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 $n$.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI S4 2013 Q2 [24]}}