2 The random variable \(X\) has probability density function \(\mathrm { f } ( x )\) where
$$\mathrm { f } ( x ) = \lambda \mathrm { e } ^ { - \lambda x } , \quad x > 0 .$$
- Obtain the moment generating function (mgf) of \(X\).
- 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 \(X\).
- Write down expressions for \(\mathrm { E } ( Y )\) and \(\operatorname { Var } ( Y )\).
Obtain the \(\operatorname { mgf }\) of \(Y\).
- Find the \(\operatorname { mgf }\) of \(Z\) where \(Z = \frac { Y - \frac { n } { \lambda } } { \frac { \sqrt { n } } { \lambda } }\).
- 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.