4. The random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by
$$\mathrm { f } ( x ) = \left\{ \begin{array} { c c }
3 k & 0 \leqslant x < 1
k x ( 4 - x ) & 1 \leqslant x \leqslant 4
0 & \text { otherwise }
\end{array} \right.$$
where \(k\) is a constant.
- Sketch f (x).
- Write down the mode of \(X\).
Given that \(\mathrm { E } ( X ) = \frac { 29 } { 16 }\)
- describe, giving a reason, the skewness of the distribution.
- Use integration to find the value of \(k\).
- Write down the lower quartile of \(X\).
Given also that \(\mathrm { P } ( 2 < X < 3 ) = \frac { 11 } { 36 }\)
- find the exact value of \(\mathrm { P } ( X > 3 )\).