6 A continuous random variable \(X\) has probability density function
$$\mathrm { f } ( x ) = \begin{cases} 4 x \mathrm { e } ^ { - 2 x } & x \geqslant 0 \\ 0 & \text { otherwise } . \end{cases}$$
- Show that the moment generating function \(\mathrm { M } _ { X } ( t )\) of \(X\) is \(\frac { 4 } { ( 2 - t ) ^ { 2 } }\). You may assume that \(x \mathrm { e } ^ { - k x } \rightarrow 0\) as \(x \rightarrow + \infty\).
- What condition on \(t\) is needed in finding \(\mathrm { M } _ { X } ( t )\) ?
- \(Y\) is the sum of three independent observations of \(X\). Find the moment generating function of \(Y\), and use your answer to find \(\operatorname { Var } ( Y )\).