2 The random variable $X$ has probability density function $\mathrm { f } ( x )$ where

$$\mathrm { f } ( x ) = \lambda \mathrm { e } ^ { - \lambda x } , \quad x > 0 .$$

(i) Obtain the moment generating function (mgf) of $X$.\\
(ii) Use the mgf to find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.

The random variable $Y$ is defined as follows:

$$Y = X _ { 1 } + \ldots + X _ { n } ,$$

where the $X _ { i }$ are independently and identically distributed as $X$.\\
(iii) Write down expressions for $\mathrm { E } ( Y )$ and $\operatorname { Var } ( Y )$.

Obtain the $\operatorname { mgf }$ of $Y$.\\
(iv) Find the $\operatorname { mgf }$ of $Z$ where $Z = \frac { Y - \frac { n } { \lambda } } { \frac { \sqrt { n } } { \lambda } }$.\\
(v) By considering the logarithm of the mgf of $Z$, show that the distribution of $Z$ tends to the standard Normal distribution as $n$ tends to infinity.

\hfill \mbox{\textit{OCR MEI S4 2016 Q2 [24]}}