8 The continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < - 1
\frac { x + 1 } { k + 1 } & - 1 \leqslant x \leqslant k
1 & x > k \end{array} \right.$$ where \(k\) is a positive constant.

1. Find, in terms of \(k\), an expression for \(\mathrm { P } ( X < 0 )\).
2. Determine an expression, in terms of \(k\), for the lower quartile, \(q _ { 1 }\).
3. Show that the probability density function of \(X\) is defined by $$\mathrm { f } ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { k + 1 } & - 1 \leqslant x \leqslant k
   0 & \text { otherwise } \end{array} \right.$$
4. Given that \(k = 11\) :
   1. sketch the graph of f;
   2. determine \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\);
   3. show that \(\mathrm { P } \left( q _ { 1 } < X < \mathrm { E } ( X ) \right) = 0.25\).