S3 June 2014 Q4
4. The random variable \(A\) is defined as
\[A = B + 4C - 3D\]where \(B\), \(C\) and \(D\) are independent random variables with
\[B \sim \mathrm{N}(6, 2^2) \qquad C \sim \mathrm{N}(7, 3^2) \qquad D \sim \mathrm{N}(4, 1.5^2)\]Find \(\mathrm{P}(A \lt 45)\) (6)
| Scheme | Marks |
|---|---|
| \(\mathrm{E}(A) = \mathrm{E}(B) + 4\mathrm{E}(C) - 3\mathrm{E}(D)\) | M1 |
| \(= 22\) | A1 |
| \(\mathrm{Var}(A) = \mathrm{Var}(B) + 16\mathrm{Var}(C) + 9\mathrm{Var}(D)\) | M1 |
| \(= 168.25\) | A1 |
| \(\mathrm{P}(A \lt 45) = \mathrm{P}\left(Z \lt \dfrac{45 - 22}{\sqrt{168.25}}\right)\) | M1 |
| \(= \mathrm{P}(Z \lt 1.773)\) \(= 0.9616\) awrt 0.962 | A1 |
| (6 marks) |
Notes
1st M1 for \(\mathrm{E}(4C) = 4\mathrm{E}(C)\) and \(-\mathrm{E}(3D) = -3\mathrm{E}(D)\)
1st A1 for 22 cao
2nd M1 for use of \(\mathrm{Var}(aX) = a^2\mathrm{Var}\,X\) and + their ‘\(9\mathrm{Var}(D)\)’
2nd A1 for 168.25 cao
3rd M1 for standardising using their mean and their sd
3rd A1 for awrt 0.962. NB Calculator gives 0.961899….