4 A discrete random variable \(X\) has the probability distribution $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l } \frac { 3 x } { 40 } & x = 1,2,3,4
\frac { x } { 20 } & x = 5
0 & \text { otherwise } \end{array} \right.$$

1. Calculate \(\mathrm { E } ( X )\).
2. Show that:
   1. \(\quad \mathrm { E } \left( \frac { 1 } { X } \right) = \frac { 7 } { 20 }\);
      (2 marks)
   2. \(\operatorname { Var } \left( \frac { 1 } { X } \right) = \frac { 7 } { 160 }\).
3. The discrete random variable \(Y\) is such that \(Y = \frac { 40 } { X }\). Calculate:
   1. \(\mathrm { P } ( Y < 20 )\);
   2. \(\mathrm { P } ( X < 4 \mid Y < 20 )\).