Conditional probability with PDF

1. The continuous random variable \(G\) has probability density function \(\mathrm { f } ( \mathrm { g } )\) given by

$$f ( g ) = \begin{cases} \frac { 1 } { 15 } ( g + 3 ) & - 1 < g \leqslant 2 \\ \frac { 3 } { 20 } & 2 < g \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$

1. Sketch the graph of \(\mathrm { f } ( \mathrm { g } )\)
2. 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 \(\operatorname { Var } ( H ) = 2.4\)
3. Find \(\mathrm { E } \left( 2 H ^ { 2 } + 3 G + 3 \right)\) Show your working clearly.
   (Solutions relying on calculator technology are not acceptable.)

4 The continuous random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \begin{cases} \frac { k } { x ^ { n } } & x \geqslant 1 \\ 0 & \text { otherwise } \end{cases}$$ where \(n\) and \(k\) are constants and \(n\) is an integer greater than 1 .

1. Find \(k\) in terms of \(n\).
2. 1. When \(n = 4\), find the cumulative distribution function of \(X\).
   2. Hence determine \(\mathrm { P } ( X > 7 \mid X > 5 )\) when \(n = 4\).
3. Determine the values of \(n\) for which \(\operatorname { Var } ( X )\) is not defined.