Expectation of reciprocals and nonlinear functions

5 A discrete random variable \(X\) has the probability distribution $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l } \frac { x } { 20 } & x = 1,2,3,4,5
\frac { x } { 24 } & x = 6
0 & \text { otherwise } \end{array} \right.$$

1. Calculate \(\mathrm { P } ( X \geqslant 5 )\).
2. 1. Show that \(\mathrm { E } \left( \frac { 1 } { X } \right) = \frac { 7 } { 24 }\).
   2. Hence, or otherwise, show that \(\operatorname { Var } \left( \frac { 1 } { X } \right) = 0.036\), correct to three decimal places.
3. Calculate the mean and the variance of \(A\), the area of rectangles having sides of length \(X + 3\) and \(\frac { 1 } { X }\).

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. \(\mathrm { P } ( X < 4 \mid Y < 20 )\).

6 The random variable \(T\) can take the value \(T = - 2\) or any value in the range \(0 \leq T < 12\) The distribution of \(T\) is given by \(\mathrm { P } ( T = - 2 ) = c , \mathrm { P } ( 0 \leq T \leq t ) = 225 k - k ( 15 - t ) ^ { 2 }\) 6

1. 1. Show that \(1 - c = 216 k\)
      [0pt] [3 marks] 6
2. (ii) Given that \(c = 0.1\), find the value of \(\mathrm { E } ( T )\)
   [0pt] [3 marks]
   6
3. Show that \(\mathrm { E } ( \sqrt { | T | } ) = \frac { 5 \sqrt { 2 } + 52 \sqrt { 3 } } { 50 }\)
   [0pt] [3 marks]