6. A random variable \(X\) has a probability density function given by
$$\begin{array} { l l }
\mathrm { f } ( x ) = \frac { 4 x ^ { 2 } ( 3 - x ) } { 27 } & 0 \leq x \leq 3
\mathrm { f } ( x ) = 0 & \text { otherwise. }
\end{array}$$
- Find the mode of \(X\).
- Find the mean of \(X\).
- Specify completely the cumulative distribution function of \(X\).
- Deduce that the median, \(m\), of \(X\) satisfies the equation \(m ^ { 4 } - 4 m ^ { 3 } + 13 \cdot 5 = 0\), and hence show that \(1.84 < m < 1.85\).
- What do these results suggest about the skewness of the distribution?