S1 June 2006 Q4
4. The random variable \(X\) has the discrete uniform distribution
\[\mathrm{P}(X = x) = \frac{1}{5}, \qquad x = 1, 2, 3, 4, 5.\](a) Write down the value of \(\mathrm{E}(X)\) and show that \(\mathrm{Var}(X) = 2\). (3)
Find
(b) \(\mathrm{E}(3X - 2)\), (2)
(c) \(\mathrm{Var}(4 - 3X)\). (2)
| Scheme | Marks |
|---|---|
| \(\mathrm{E}(X) = 3;\) | B1 |
| \(\mathrm{Var}(X) = \dfrac{25 - 1}{12} = 2\) **AG** \(\mathrm{Var}(X) = 1^2 \times \dfrac{1}{5} + 2^2 \times \dfrac{1}{5} + 3^2 \times \dfrac{1}{5} \ldots - 3^2 = 11 - 9 = 2\) **AG** | M1A1 |
| (3) |
Notes
Accept (55/5)-9 as minimum evidence.
| Scheme | Marks |
|---|---|
| \(\mathrm{E}(3X - 2) = 3\mathrm{E}(X) - 2 = 7\) | M1A1ft |
| (2) |
| Scheme | Marks |
|---|---|
| \(\mathrm{Var}(4 - 3x) = 3^2\,\mathrm{Var}(X) = 18\) | M1A1 |
| (2) | |
| (7 marks) |