4 The continuous random variable \(X\) has probability density function
$$f ( x ) = \left\{ \begin{array} { c c }
x & 0 \leqslant x \leqslant 1
2 - x & 1 \leqslant x \leqslant 2
0 & \text { otherwise }
\end{array} \right.$$
- Show that the moment generating function of \(X\) is \(\frac { \left( \mathrm { e } ^ { t } - 1 \right) ^ { 2 } } { t ^ { 2 } }\).
\(Y _ { 1 }\) and \(Y _ { 2 }\) are independent observations of a random variable \(Y\). The moment generating function of \(Y _ { 1 } + Y _ { 2 }\) is \(\frac { \left( \mathrm { e } ^ { t } - 1 \right) ^ { 2 } } { t ^ { 2 } }\). - Write down the moment generating function of \(Y\).
- Use the expansion of \(\mathrm { e } ^ { t }\) to find \(\operatorname { Var } ( Y )\).
- Deduce the value of \(\operatorname { Var } ( X )\).