S2 January 2005 Q1
1. The random variables \(R\), \(S\) and \(T\) are distributed as follows\[R \sim \mathrm{B}(15, 0.3), \quad S \sim \mathrm{Po}(7.5), \quad T \sim \mathrm{N}(8, 2^2).\]Find
(a) \(\mathrm{P}(R = 5)\), (2)
(b) \(\mathrm{P}(S = 5)\), (1)
(c) \(\mathrm{P}(T = 5)\). (1)
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(R = 5) = \mathrm{P}(R \leqslant 5) - \mathrm{P}(R \leqslant 4) = 0.7216 - 0.5155\) | M1 |
| \(= \underline{0.2061}\) | A1 |
| (2) |
Notes
M1 can be implied
A1 awrt 0.2061
(or: \({}^{15}\mathrm{C}_5(0.3)^5(0.7)^{10} = 0.206130\ldots\))
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(S = 5) = 0.2414 - 0.1321 = \underline{0.1093}\) | B1 |
| (1) |
Notes
B1 accept awrt 0.1093 or awrt 0.1094
(or: \(\dfrac{7.5^5\mathrm{e}^{-7.5}}{5!} = 0.10937459\ldots\))
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(T = 5) = 0\) | B1 cao |
| (1) | |
| (4 marks) |