2 The continuous random variable \(X\) has probability density function given by
$$f ( x ) = \begin{cases} 4 x e ^ { - 2 x } & x \geqslant 0
0 & \text { otherwise } \end{cases}$$
- Show that the moment generating function ( mgf ) of \(X\) is
$$\frac { 4 } { ( 2 - t ) ^ { 2 } } , \text { where } | t | < 2$$
- Explain why the \(\operatorname { mgf }\) of \(- X\) is \(\frac { 4 } { ( 2 + t ) ^ { 2 } }\).
- Two random observations of \(X\) are denoted by \(X _ { 1 }\) and \(X _ { 2 }\). What is the \(\operatorname { mgf }\) of \(X _ { 1 } - X _ { 2 }\) ?