6. A continuous random variable \(X\) has probability density function
$$f ( x ) = \begin{cases} a x ^ { 2 } + b x & 1 \leqslant x \leqslant 7
0 & \text { otherwise } \end{cases}$$
where \(a\) and \(b\) are constants.
- Show that \(114 a + 24 b = 1\)
Given that \(a = \frac { 1 } { 90 }\)
- use algebraic integration to find \(\mathrm { E } ( X )\)
- find the cumulative distribution function of \(X\), specifying it for all values of \(x\)
- find \(\mathrm { P } ( X > \mathrm { E } ( X ) )\)
- use your answer to part (d) to describe the skewness of the distribution.