| Exam Board | OCR |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2010 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Sum or total of normal variables |
| Difficulty | Moderate -0.3 This is a straightforward application of standard results for sums of independent normal variables. Part (i) requires recognizing Y as the sum of 4 independent normals and standardizing. Parts (ii)-(iii) test understanding that Y and X are independent, requiring calculation of E(V) and Var(V) using linearity properties. All steps are routine for S3 level with no novel insight required, making it slightly easier than average. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.04b Linear combinations: of normal distributions |
2 The amount of tomato juice, $X \mathrm { ml }$, dispensed into cartons of a particular brand has a normal distribution with mean 504 and standard deviation 3 . The juice is sold in packs of 4 cartons, filled independently. The total amount of juice in one pack is $Y \mathrm { ml }$.\\
(i) Find $\mathrm { P } ( Y < 2000 )$.
The random variable $V$ is defined as $Y - 4 X$.\\
(ii) Find $\mathrm { E } ( V )$ and $\operatorname { Var } ( V )$.\\
(iii) What is the probability that the amount of juice in a randomly chosen pack is more than 4 times the amount of juice in a randomly chosen carton?
\hfill \mbox{\textit{OCR S3 2010 Q2 [8]}}