| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2020 |
| Session | Specimen |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Multiple stage process probability |
| Difficulty | Standard +0.8 This question requires understanding of linear combinations of independent normal random variables, including finding distributions of sums and differences with coefficients. Part (b) involves comparing 3Y - X which requires careful handling of variance scaling. This is a standard S2 topic but requires multiple steps and proper application of variance rules, making it moderately challenging but not exceptional for Further Maths Statistics. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(N(5100,\ 5 \times 45^2)\) or \(N(5100, 10125)\) | 1, B1 | Seen or implied |
| \(\dfrac{5200 - 5100'}{\sqrt{10125'}}\) \((= 0.994)\) | 1, M1 | Standardising with their new mean and new variance |
| \(\Phi(0.994')\) | 1, M1 | Area consistent with their working with normal |
| \(= 0.840\) (3 sf) | 1, A1 | |
| Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use of \(E - 3L\) or similar | 1, M1 | |
| \(E(E - 3L) = -260\) | 1, B1 | or \(2800 - 3 \times 1020\) |
| \(\text{Var}(E - 3L) = 52^2 + 9 \times 45^2\) or \(20929\) | 1, B1 | Using a positive variance with \(45^2\) and \(52^2\) combined |
| \(\dfrac{0 - (-260')}{\sqrt{20929'}}\) \((= 1.797)\) | 1, M1 | |
| \(1 - \Phi(1.797')\) | 1, M1 | Consistent area, must clearly be \(\Phi\) |
| \(= 0.0361\) (3 sf) or \(0.0362\) | 1, A1 | |
| Total: 6 |
## Question 4(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $N(5100,\ 5 \times 45^2)$ or $N(5100, 10125)$ | 1, B1 | Seen or implied |
| $\dfrac{5200 - 5100'}{\sqrt{10125'}}$ $(= 0.994)$ | 1, M1 | Standardising with their new mean and new variance |
| $\Phi(0.994')$ | 1, M1 | Area consistent with their working with normal |
| $= 0.840$ (3 sf) | 1, A1 | |
| **Total: 4** | | |
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## Question 4(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use of $E - 3L$ or similar | 1, M1 | |
| $E(E - 3L) = -260$ | 1, B1 | or $2800 - 3 \times 1020$ |
| $\text{Var}(E - 3L) = 52^2 + 9 \times 45^2$ or $20929$ | 1, B1 | Using a positive variance with $45^2$ and $52^2$ combined |
| $\dfrac{0 - (-260')}{\sqrt{20929'}}$ $(= 1.797)$ | 1, M1 | |
| $1 - \Phi(1.797')$ | 1, M1 | Consistent area, must clearly be $\Phi$ |
| $= 0.0361$ (3 sf) or $0.0362$ | 1, A1 | |
| **Total: 6** | | |4 The lifetimes, in hours, of Longlive light bulbs and Enerlow light bulbs have the independent distributions $\mathrm { N } \left( 1020,45 ^ { 2 } \right)$ and $\mathrm { N } \left( 2800,52 ^ { 2 } \right)$ respectively.\\
(a) Find the probability that the total of the lifetimes of five randomly chosen Longlive bulbs is less than 5200 hours.\\
(b) Find the probability that the lifetime of a randomly chosen Enerlow bulb is at least three times that of a randomly chosen Longlive bulb.\\
\hfill \mbox{\textit{CAIE S2 2020 Q4 [10]}}