7. The random variable \(X\) has probability density function
$$\mathrm { f } ( x ) = \begin{cases} k \left( - x ^ { 2 } + 5 x - 4 \right) , & 1 \leq x \leq 4
0 , & \text { otherwise } \end{cases}$$
- Show that \(k = \frac { 2 } { 9 }\).
Find
- \(\mathrm { E } ( X )\),
- the mode of \(X\).
- the cumulative distribution function \(\mathrm { F } ( x )\) for all \(x\).
- Evaluate \(\mathrm { P } ( X \leq 2.5 )\),
- Deduce the value of the median and comment on the shape of the distribution.