7 The continuous random variable \(X\) has probability density function given by
$$f ( x ) = \begin{cases} \frac { 3 } { 4 } \left( x ^ { 2 } - 1 \right) & 1 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$
- Sketch the probability density function of \(X\).
- Show that the mean, \(\mu\), of \(X\) is 1.6875 .
- Show that the standard deviation, \(\sigma\), of \(X\) is 0.2288 , correct to 4 decimal places.
- Find \(\mathrm { P } ( 1 \leqslant X \leqslant \mu + \sigma )\).