| Exam Board | OCR |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2010 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Random Variables |
| Type | Covariance calculation and independence |
| Difficulty | Standard +0.3 This is a straightforward application of variance and covariance formulas. Students need to expand Var(2A - 3B) = 4Var(A) - 12Cov(A,B) + 9Var(B), substitute given values, and solve for Cov(A,B). Part (ii) is immediate recall that independence requires Cov = 0. Slightly above average difficulty due to being S4 (Further Maths) content, but the algebraic manipulation is routine. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\text{Var}(2A-3B) = 4\text{Var}(A) + 9\text{Var}(B) - 12\text{Cov}(A,B)\) | M1 | Correct formula. Allow one error |
| \(\Rightarrow 18 = 36 + 54 - 12\text{Cov}(A,B)\) | A1 | Substitute relevant values |
| \(\Rightarrow \text{Cov}(A,B) = 6\) | A1 [3] | CAO |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Since \(\text{Cov}(A,B) \neq 0\), \(A\) and \(B\) are not independent | B1ft [1] | Must have a reason. ft \(\text{Cov} \neq 0\) |
## Question 1:
### Part (i)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\text{Var}(2A-3B) = 4\text{Var}(A) + 9\text{Var}(B) - 12\text{Cov}(A,B)$ | M1 | Correct formula. Allow one error |
| $\Rightarrow 18 = 36 + 54 - 12\text{Cov}(A,B)$ | A1 | Substitute relevant values |
| $\Rightarrow \text{Cov}(A,B) = 6$ | A1 **[3]** | CAO |
### Part (ii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Since $\text{Cov}(A,B) \neq 0$, $A$ and $B$ are not independent | B1ft **[1]** | Must have a reason. ft $\text{Cov} \neq 0$ |
---1 For the variables $A$ and $B$, it is given that $\operatorname { Var } ( A ) = 9 , \operatorname { Var } ( B ) = 6$ and $\operatorname { Var } ( 2 A - 3 B ) = 18$.\\
(i) Find $\operatorname { Cov } ( A , B )$.\\
(ii) State with a reason whether $A$ and $B$ are independent.
\hfill \mbox{\textit{OCR S4 2010 Q1 [4]}}