6. A continuous random variable $X$ has probability density function $\mathrm { f } ( x )$ where

$$f ( x ) = \begin{cases} k \left( 4 x - x ^ { 3 } \right) , & 0 \leqslant x \leqslant 2 \\ 0 , & \text { otherwise } \end{cases}$$

where $k$ is a positive integer.\\
(a) Show that $k = \frac { 1 } { 4 }$.

Find\\
(b) $\mathrm { E } ( X )$,\\
(c) the mode of $X$,\\
(d) the median of $X$.\\
(e) Comment on the skewness of the distribution.\\
(f) Sketch f(x).\\

\hfill \mbox{\textit{Edexcel S2  Q6}}