7 A continuous random variable \(X _ { 1 }\) has probability density function given by
$$f ( x ) = \begin{cases} k x & 0 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$
where \(k\) is a constant.
- Show that \(k = \frac { 1 } { 2 }\).
- Sketch the graph of \(y = \mathrm { f } ( x )\).
- Find \(\mathrm { E } \left( X _ { 1 } \right)\) and \(\operatorname { Var } \left( X _ { 1 } \right)\).
- Sketch the graph of \(y = \mathrm { f } ( x - 1 )\).
- The continuous random variable \(X _ { 2 }\) has probability density function \(\mathrm { f } ( x - 1 )\) for all \(x\). Write down the values of \(\mathrm { E } \left( X _ { 2 } \right)\) and \(\operatorname { Var } \left( X _ { 2 } \right)\).