| Exam Board | OCR |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2013 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Pure expectation and variance calculation |
| Difficulty | Standard +0.3 This is a straightforward application of expectation and variance rules for linear combinations. Part (i) requires routine algebraic manipulation using E(aX+bY+c) and Var(aX+bY) formulas with independent variables. Part (ii) is a simple normal probability calculation once parameters are found. Slightly above average due to the Poisson component and algebraic setup, but still a standard textbook exercise requiring no novel insight. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(E(S) = 50 - 4 + c = 408 \Rightarrow c = 362\) | M1, A1 | Using \(E(aX+bY+c)\) |
| \(\text{Var}(S) = 25\sigma^2 + 8 = 408 \Rightarrow \sigma^2 = 16\) AG | M1, A1 | Using \(\text{Var}(aX+bY+c)\) |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(X \geq 2) = P(Z \geq -8/4) = 0.9772\) | M1, A1 | \(1 - \Phi(-2)\) |
| [2] |
# Question 1:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $E(S) = 50 - 4 + c = 408 \Rightarrow c = 362$ | M1, A1 | Using $E(aX+bY+c)$ |
| $\text{Var}(S) = 25\sigma^2 + 8 = 408 \Rightarrow \sigma^2 = 16$ AG | M1, A1 | Using $\text{Var}(aX+bY+c)$ |
| | **[4]** | |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(X \geq 2) = P(Z \geq -8/4) = 0.9772$ | M1, A1 | $1 - \Phi(-2)$ |
| | **[2]** | |
---1 The independent random variables $X$ and $Y$ have the distributions $\mathrm { N } \left( 10 , \sigma ^ { 2 } \right)$ and $\operatorname { Po } ( 2 )$ respectively. The random variable $S$ is given by $S = 5 X - 2 Y + c$, where $c$ is a constant.\\
It is given that $\mathrm { E } ( S ) = \operatorname { Var } ( S ) = 408$.\\
(i) Find the value of $c$ and show that $\sigma ^ { 2 } = 16$.\\
(ii) Find $\mathrm { P } ( X \geqslant \mathrm { E } ( Y ) )$.
\hfill \mbox{\textit{OCR S3 2013 Q1 [6]}}