3 The continuous random variable \(X\) has a cumulative distribution function defined by $$\mathrm { F } ( x ) = \left\{ \begin{array} { c l } 0 & x < - 5
\frac { x + 5 } { 20 } & - 5 \leqslant x \leqslant 15
1 & x > 15 \end{array} \right.$$

1. Show that, for \(- 5 \leqslant x \leqslant 15\), the probability density function, \(\mathrm { f } ( x )\), of \(X\) is given by \(\mathrm { f } ( x ) = \frac { 1 } { 20 }\).
   (1 mark)
2. Find:
   1. \(\mathrm { P } ( X \geqslant 7 )\);
   2. \(\mathrm { P } ( X \neq 7 )\);
   3. \(\mathrm { E } ( X )\);
   4. \(\mathrm { E } \left( 3 X ^ { 2 } \right)\).