- The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by
$$f ( x ) = \begin{cases} k \left( x ^ { 2 } - 2 x + 2 \right) & 0 < x \leqslant 3
3 k & 3 < x \leqslant 4
0 & \text { otherwise } \end{cases}$$
where \(k\) is a constant.
- Show that \(k = \frac { 1 } { 9 }\)
- Find the cumulative distribution function \(\mathrm { F } ( x )\).
- Find the mean of \(X\).
- Show that the median of \(X\) lies between \(x = 2.6\) and \(x = 2.7\)