A2 June 2022 Q3
3. The random variable \(X \sim \mathrm{N}(5,\ 0.4^2)\) and the random variable \(Y \sim \mathrm{N}(8,\ 0.1^2)\)
\(X\) and \(Y\) are independent random variables.
A random sample of \(a\) independent observations is taken from the distribution of \(X\) and one observation is taken from the distribution of \(Y\)
The random variable \(W = X_1 + X_2 + X_3 + \ldots + X_a + bY\) and has the distribution \(\mathrm{N}(169,\ 2^2)\)
Find the value of \(a\) and the value of \(b\) (6)
| Scheme | Marks | AO |
|---|---|---|
| \(5a + 8b = 169\) | B1 | 1.1b |
| \(0.16a + 0.01b^2 = 4\) | B1 | 1.1b |
| \(a = 33.8 - 1.6b \rightarrow 0.16(33.8 - 1.6b) + 0.01b^2 = 4\) | M1 | 2.1 |
| \(0.01b^2 - 0.256b + 1.408 = 0 \rightarrow b = \dfrac{0.256 \pm \sqrt{0.256^2 - 4(0.01)(1.408)}}{2(0.01)}\) | M1 | 1.1b |
| \(b = 8,\ a = 21\) (reject \(b = 17.6,\ a = 5.64\) since \(a \in \mathbb{Z}^+\)) | A1A1 | 1.1b 2.2a |
| (6 marks) |
Notes
B1: Correct equation for the means
B1: Correct equation for the variances (allow \(0.4^2 a + 0.1^2 b^2 = 4\))
M1: Attempt to solve two simultaneous equations in \(a\) and \(b\) by eliminating one variable
M1: Attempt to solve their quadratic (must be seen if answers are incorrect)
A1: \(b = 8\) or \(a = 21\) or both sets of values of \(a\) and \(b\) without rejecting
A1: Choosing correct pair of solutions \(b = 8,\ a = 21\) only