5 The random variable \(X\) has a Poisson distribution with mean \(\lambda\). It is given that the moment generating function of \(X\) is \(e ^ { \lambda \left( e ^ { t } - 1 \right) }\).
- Use the moment generating function to verify that the mean of \(X\) is \(\lambda\), and to show that the variance of \(X\) is also \(\lambda\).
- Five independent observations of \(X\) are added to produce a new variable \(Y\). Find the moment generating function of \(Y\), simplifying your answer.