5 The discrete random variable \(R\) has the following probability distribution.
| \(\boldsymbol { r }\) | 1 | 2 | 4 |
| \(\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )\) | \(\frac { 1 } { 4 }\) | \(\frac { 1 } { 2 }\) | \(\frac { 1 } { 4 }\) |
- Calculate exact values for \(\mathrm { E } ( R )\) and \(\operatorname { Var } ( R )\).
- By tabulating the probability distribution for \(X = \frac { 1 } { R ^ { 2 } }\), show that \(\mathrm { E } ( X ) = \frac { 25 } { 64 }\).
- Hence find the value of the mean of the area of a rectangle which has sides of length \(\frac { 8 } { R }\) and \(\left( R + \frac { 8 } { R } \right)\).
(3 marks)