- A random variable \(X\) has probability density function given by
$$f ( x ) = \left\{ \begin{array} { c l }
- \frac { 2 } { 9 } x + \frac { 8 } { 9 } & 1 \leqslant x \leqslant 4
0 & \text { otherwise }
\end{array} \right.$$
- Show that the cumulative distribution function \(\mathrm { F } ( x )\) can be written in the form \(a x ^ { 2 } + b x + c\), for \(1 \leqslant x \leqslant 4\) where \(a , b\) and \(c\) are constants.
- Define fully the cumulative distribution function \(\mathrm { F } ( x )\).
- Show that the upper quartile of \(X\) is 2.5 and find the lower quartile.
Given that the median of \(X\) is 1.88
- describe the skewness of the distribution. Give a reason for your answer.