- The continuous random variable \(X\) has probability density function given by
$$f ( x ) = \left\{ \begin{array} { c c }
\frac { 3 } { 32 } ( x - 1 ) ( 5 - x ) & 1 \leqslant x \leqslant 5
0 & \text { otherwise }
\end{array} \right.$$
- Sketch \(\mathrm { f } ( x )\) showing clearly the points where it meets the \(x\)-axis.
- Write down the value of the mean, \(\mu\), of \(X\).
- Show that \(\mathrm { E } \left( X ^ { 2 } \right) = 9.8\)
- Find the standard deviation, \(\sigma\), of \(X\).
The cumulative distribution function of \(X\) is given by
$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c }
0 & x < 1
\frac { 1 } { 32 } \left( a - 15 x + 9 x ^ { 2 } - x ^ { 3 } \right) & 1 \leqslant x \leqslant 5
1 & x > 5
\end{array} \right.$$
where \(a\) is a constant. - Find the value of \(a\).
- Show that the lower quartile of \(X , q _ { 1 }\), lies between 2.29 and 2.31
- Hence find the upper quartile of \(X\), giving your answer to 1 decimal place.
- Find, to 2 decimal places, the value of \(k\) so that
$$\mathrm { P } ( \mu - k \sigma < X < \mu + k \sigma ) = 0.5$$