| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2022 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Pure expectation and variance calculation |
| Difficulty | Moderate -0.3 Part (a) requires straightforward application of expectation and variance formulas for linear combinations (E(aX+bY) and Var(aX+bY) with independence), then taking square root for SD. Part (b) requires recognizing that Y=15X forces specific values, then calculating P(X=0,Y=0) + P(X=1,Y=15) using Poisson and Binomial pmfs. While multi-step, these are standard S2 techniques with no novel insight required, making it slightly easier than average A-level. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities5.02i Poisson distribution: random events model5.02m Poisson: mean = variance = lambda5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(E(Y) = \frac{20}{4} = 5\), \(\text{Var}(Y) = 20 \times \frac{1}{4} \times \frac{3}{4} = \frac{15}{4}\) | B1 | Both; OE, SOI |
| \(\text{Var}(X) = 2\) | B1 | SOI or standard deviation \(= \sqrt{2}\) |
| \(E(X - 3Y) = -13\) | B1 | |
| \(\text{Var}(X - 3Y) = 2 + 9 \times \frac{15}{4} = 35.75\) | M1 | Correct formula using *their* values |
| Standard deviation of \((X-3Y) = 5.98\) (3 s.f.) or \(\frac{1}{2}\sqrt{143}\) | A1 | CWO |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \((0,0)\) and \((1,15)\) | M1 | |
| \(e^{-2} \times \left(\frac{3}{4}\right)^{20} + e^{-2} \times 2 \times \binom{20}{15}\left(\frac{3}{4}\right)^5\left(\frac{1}{4}\right)^{15}\) | M1 | |
| \(0.000430\) (3 sf) | A1 | CWO (must have evidence of addition); Allow 0.00043 or \(4.3(0) \times 10^{-4}\) |
## Question 4(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $E(Y) = \frac{20}{4} = 5$, $\text{Var}(Y) = 20 \times \frac{1}{4} \times \frac{3}{4} = \frac{15}{4}$ | B1 | Both; OE, SOI |
| $\text{Var}(X) = 2$ | B1 | SOI or standard deviation $= \sqrt{2}$ |
| $E(X - 3Y) = -13$ | B1 | |
| $\text{Var}(X - 3Y) = 2 + 9 \times \frac{15}{4} = 35.75$ | M1 | Correct formula using *their* values |
| Standard deviation of $(X-3Y) = 5.98$ (3 s.f.) or $\frac{1}{2}\sqrt{143}$ | A1 | CWO |
---
## Question 4(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $(0,0)$ and $(1,15)$ | M1 | |
| $e^{-2} \times \left(\frac{3}{4}\right)^{20} + e^{-2} \times 2 \times \binom{20}{15}\left(\frac{3}{4}\right)^5\left(\frac{1}{4}\right)^{15}$ | M1 | |
| $0.000430$ (3 sf) | A1 | CWO (must have evidence of addition); Allow 0.00043 or $4.3(0) \times 10^{-4}$ |
---4 The independent random variables $X$ and $Y$ have distributions $\operatorname { Po } ( 2 )$ and $\mathrm { B } \left( 20 , \frac { 1 } { 4 } \right)$ respectively.
\begin{enumerate}[label=(\alph*)]
\item Find the mean and standard deviation of $X - 3 Y$.
\item Find $\mathrm { P } ( Y = 15 X )$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2022 Q4 [8]}}