6 The random variable \(X\) has probability density function defined by
$$f ( x ) = \begin{cases} \frac { 1 } { 40 } ( x + 7 ) & 1 \leqslant x \leqslant 5
0 & \text { otherwise } \end{cases}$$
- Sketch the graph of f.
- Find the exact value of \(\mathrm { E } ( X )\).
- Prove that the distribution function F , for \(1 \leqslant x \leqslant 5\), is defined by
$$\mathrm { F } ( x ) = \frac { 1 } { 80 } ( x + 15 ) ( x - 1 )$$
- Hence, or otherwise:
- find \(\mathrm { P } ( 2.5 \leqslant X \leqslant 4.5 )\);
- show that the median, \(m\), of \(X\) satisfies the equation \(m ^ { 2 } + 14 m - 55 = 0\).
- Calculate the value of the median of \(X\), giving your answer to three decimal places.