S1 January 2006 Q2
2. The random variable \(X\) has probability distribution
| \(x\) | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| \(\mathrm{P}(X = x)\) | 0.10 | \(p\) | 0.20 | \(q\) | 0.30 |
(a) Given that \(\mathrm{E}(X) = 3.5\), write down two equations involving \(p\) and \(q\). (3)
Find
(b) the value of \(p\) and the value of \(q\), (3)
(c) \(\mathrm{Var}(X)\), (4)
(d) \(\mathrm{Var}(3 - 2X)\). (2)
| Scheme | Marks |
|---|---|
| \(p + q = 0.4\) | B1 |
| \(2p + 4q = 1.3\) | M1A1 |
| (3) |
Notes
Consider with (b).
| Scheme | Marks |
|---|---|
| Attempt to solve | M1 |
| \(p = 0.15, q = 0.25\) | A1A1 |
| (3) |
Notes
If both seen, award 3.
| Scheme | Marks |
|---|---|
| \(\mathrm{E}(X^2) = 1^2 \times 0.10 + 2^2 \times 0.15 + \ldots + 5^2 \times 0.30 = 14\) | M1A1ft |
| \(\mathrm{Var}(X) = 14 - 3.5^2 = 1.75\) | M1A1 |
| (4) |
| Scheme | Marks |
|---|---|
| \(\mathrm{Var}(3 - 2X) = 4\mathrm{Var}(X) = 7.00\) | M1A1ft |
| (2) | |
| (12 marks) |