- The continuous random variable \(X\) has cumulative distribution function given by
$$F ( x ) = \left\{ \begin{array} { c r }
0 & x < 0
k \left( x - a x ^ { 2 } \right) & 0 \leqslant x \leqslant 4
1 & x > 4
\end{array} \right.$$
The values of \(a\) and \(k\) are positive constants such that \(\mathrm { P } ( X < 2 ) = \frac { 2 } { 3 }\)
- Find the exact value of the median of \(X\)
- Find the probability density function of \(X\)
- Hence, deduce the value of the mode of \(X\), giving a reason for your answer.