3 The random variable \(X\) has the following probability density function:
$$\mathrm { f } ( x ) = \begin{cases} k \left( 1 - x ^ { 2 } \right) & - 1 \leqslant x \leqslant 1
0 & \text { elsewhere } \end{cases}$$
where \(k\) is a positive constant.
- Calculate the value of \(k\).
- Sketch the probability density function.
- Calculate \(\operatorname { Var } ( X )\).
- Find a cubic equation satisfied by the upper quartile \(q\), and hence verify that \(q = 0.35\) to 2 decimal places.
- A random sample of 40 values of \(X\) is taken. Using a suitable approximating distribution, calculate the probability that the mean of these values is greater than 0.125 . Justify your choice of distribution.