6. The continuous random variable X has cumulative distribution function \(\mathrm { F } ( x )\) given by
$$\mathrm { F } ( x ) = \left\{ \begin{array} { l r }
0 , & x < 1
\frac { 1 } { 27 } \left( - x ^ { 3 } + 6 x ^ { 2 } - 5 \right) , & 1 \leq x \leq 4
1 , & x > 4
\end{array} \right.$$
- Find the probability density function \(\mathrm { f } ( x )\).
- Find the mode of \(X\).
- Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
- Find the mean \(\mu\) of X .
- Show that \(\mathrm { F } ( \mu ) > 0.5\).
- Show that the median of \(X\) lies between the mode and the mean.