| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2017 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Expectation and variance with context application |
| Difficulty | Moderate -0.3 This is a straightforward application of linear transformations of normal variables and combining independent normals. Part (i) requires only E(aX) = aE(X) and Var(aX) = a²Var(X), while part (ii) needs recognizing that aX + bY is normal with combined mean/variance, then a standard normal probability calculation. All steps are routine S2 techniques with no problem-solving 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 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Mean \(= 3.2 \times 90 = 288\) | B1 | |
| Variance \(= 0.4^2 \times 90^2\) | M1 | |
| \(= 1296\) | A1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Mean \(= \text{'288'} + 4.3 \times 95 = 696.5\) | B1 FT | |
| Variance \(= \text{'1296'} + 0.6^2 \times 95^2 = 4545\) | B1 FT | FT their (i) |
| \(\frac{670-696.5}{\sqrt{4545}}\) \((= -0.393)\) | M1 | FT Var provided both given Vars used standardising (ignore cc); no sd/Var mix |
| \(1 - \phi(\text{'-0.393'}) = \phi(\text{'0.393'})\) | M1 | correct area consistent with their working (i.e. their mean) |
| \(= 0.653\) (3 sf) | A1 | |
| Total: 5 |
## Question 6(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Mean $= 3.2 \times 90 = 288$ | B1 | |
| Variance $= 0.4^2 \times 90^2$ | M1 | |
| $= 1296$ | A1 | |
| **Total: 3** | | |
## Question 6(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Mean $= \text{'288'} + 4.3 \times 95 = 696.5$ | B1 FT | |
| Variance $= \text{'1296'} + 0.6^2 \times 95^2 = 4545$ | B1 FT | FT their (i) |
| $\frac{670-696.5}{\sqrt{4545}}$ $(= -0.393)$ | M1 | FT Var provided both given Vars used standardising (ignore cc); no sd/Var mix |
| $1 - \phi(\text{'-0.393'}) = \phi(\text{'0.393'})$ | M1 | correct area consistent with their working (i.e. their mean) |
| $= 0.653$ (3 sf) | A1 | |
| **Total: 5** | | |
---6 The numbers of barrels of oil, in millions, extracted per day in two oil fields $A$ and $B$ are modelled by the independent random variables $X$ and $Y$ respectively, where $X \sim \mathrm {~N} \left( 3.2,0.4 ^ { 2 } \right)$ and $Y \sim \mathrm {~N} \left( 4.3,0.6 ^ { 2 } \right)$. The income generated by the oil from the two fields is $\$ 90$ per barrel for $A$ and $\$ 95$ per barrel for $B$.\\
(i) Find the mean and variance of the daily income, in millions of dollars, generated by field $A$. [3]\\
(ii) Find the probability that the total income produced by the two fields in a day is at least $\$ 670$ million.\\
\hfill \mbox{\textit{CAIE S2 2017 Q6 [8]}}