A2 October 2020 Q2
2. The discrete random variables \(W\), \(X\) and \(Y\) are distributed as follows
\[W \sim \mathrm{B}(10, 0.4) \qquad\qquad X \sim \mathrm{Po}(4) \qquad\qquad Y \sim \mathrm{Po}(3)\](a) Explain whether or not \(\mathrm{Po}(4)\) would be a good approximation to \(\mathrm{B}(10, 0.4)\) (1)
(b) State the assumption required for \(X + Y\) to be distributed as \(\mathrm{Po}(7)\) (1)
Given the assumption in part (b) holds,
(c) find \(\mathrm{P}(X + Y \lt \mathrm{Var}(W))\) (2)
| Scheme | Marks | AO |
|---|---|---|
| requires large \(n\)/small \(p\) so not a good approximation | B1 | 3.5b |
| (1) |
Notes
B1: Correct reason why the model would not be appropriate and correct conclusion. Condone e.g. ‘\(p\) is close to 0.5’ for \(p\) is not small.
Mean is not equal to variance on its own in B0.
| Scheme | Marks | AO |
|---|---|---|
| \(X\) and \(Y\) must be independent | B1 | 2.4 |
| (1) |
Notes
B1: Correct explanation mentioning independence (oe).
Ignore extraneous comments.
| Scheme | Marks | AO |
|---|---|---|
| \(\mathrm{P}(X + Y \lt 2.4)\) from \(\mathrm{Po}(7)\) [\(\mathrm{P}(X + Y \leqslant 2)\)] | M1 | 3.4 |
| \(= 0.029636\ldots\) awrt 0.0296 | A1 | 1.1b |
| (2) | ||
| (4 marks) |
Notes
M1: Using \(\mathrm{Po}(7)\) with 2.4
A1: awrt 0.0296