5 Aiden takes his car to a garage for its MOT test. The probability that his car will need to have \(X\) tyres replaced is shown in the table.
| \(\boldsymbol { x }\) | 0 | 1 | 2 | 3 | 4 |
| \(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\) | 0.1 | 0.35 | 0.25 | 0.2 | 0.1 |
- Show that the mean of \(X\) is 1.85 and calculate the variance of \(X\).
- The charge for the MOT test is \(\pounds c\) and the cost of each new tyre is \(\pounds n\). The total amount that Aiden must pay the garage is \(\pounds T\).
- Express \(T\) in terms of \(c , n\) and \(X\).
- Hence, using your results from part (a), find expressions for \(\mathrm { E } ( T )\) and \(\operatorname { Var } ( T )\).