- The continuous random variable \(G\) has probability density function \(\mathrm { f } ( \mathrm { g } )\) given by
$$f ( g ) = \begin{cases} \frac { 1 } { 15 } ( g + 3 ) & - 1 < g \leqslant 2
\frac { 3 } { 20 } & 2 < g \leqslant 4
0 & \text { otherwise } \end{cases}$$
- Sketch the graph of \(\mathrm { f } ( \mathrm { g } )\)
- Find \(\mathrm { P } ( ( 1 \leqslant 2 G \leqslant 6 ) \mid G \leqslant 2 )\)
The continuous random variable \(H\) is such that \(\mathrm { E } ( H ) = 12\) and \(\operatorname { Var } ( H ) = 2.4\)
- Find \(\mathrm { E } \left( 2 H ^ { 2 } + 3 G + 3 \right)\)
Show your working clearly.
(Solutions relying on calculator technology are not acceptable.)