5 A continuous random variable \(X\) has probability density function
$$f ( x ) = \begin{cases} \frac { 1 } { 2 } \pi \sin ( \pi x ) & 0 \leqslant x \leqslant 1
0 & \text { otherwise } \end{cases}$$
- Show that this is a valid probability density function. [4]
- Sketch the curve \(\boldsymbol { y } = \mathbf { f } ( \boldsymbol { x } )\) and write down the value of \(\mathbf { E } \boldsymbol { ( } \boldsymbol { X } \boldsymbol { ) }\). [3]
- Find the value \(q\) such that \(\mathrm { P } ( X > q ) = 0.75\). [3]
- Write down an expression, including an integral, for \(\operatorname { Var } ( X )\). (Do not attempt to evaluate the integral.) [2]
- A student states that " \(X\) is more likely to occur when \(x\) is close to \(\mathrm { E } ( X )\)." Give an improved version of this statement. [1]