| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Session | Specimen |
| Marks | 1 |
| Topic | Linear combinations of normal random variables |
| Type | Two or more different variables |
| Difficulty | Standard +0.8 This requires understanding that catching the bus means walk time < (bus arrival - 0845), forming the inequality W < B - 45, then recognizing this as B - W > 45 where B - W is a linear combination of normals. Students must correctly identify parameters (mean = 60 - 10 = 50, variance = 1² + 1.5² = 3.25) and standardize. The conceptual leap to reformulate as a difference of independent normals elevates this above routine normal distribution questions. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation5.04b Linear combinations: of normal distributions |
**Question 7**
$X \sim \mathrm{N}(0855,\ 1.5^2)$, $\ Y \sim \mathrm{N}(0900,\ 1^2)$
$Y - X \sim \mathrm{N}(5,\ 1.5^2 + 1^2)$ B1B1
$$z = \frac{0-5}{\sqrt{3.25}} = -2.7735$$ M1A1A1
$\mathrm{P}(Y - X > 0) = 0.997$ A1 **[6 marks]**7 The time taken for me to walk from my house to the bus stop has a normal distribution with mean 10 minutes and standard deviation 1.5 minutes. The arrival time of the bus is normally distributed with mean 0900 and standard deviation 1 minute. If the bus arrives early it does not wait. I leave home at 0845 . Find, correct to 3 decimal places, the probability that I catch the bus.
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 Q7 [1]}}