| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2016 |
| Session | March |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Two independent Poisson sums |
| Difficulty | Standard +0.3 This is a straightforward application of Poisson distribution with rate scaling and normal approximation. Part (i) requires basic Poisson calculation with rate adjustment, part (ii) is standard normal approximation to Poisson (λ=40.15 is large enough), and part (iii) uses the additive property of independent Poisson variables. All techniques are routine for S2 level with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04d Normal approximation to binomial2.04e Normal distribution: as model N(mu, sigma^2)5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\lambda = 3.3 \times \frac{25}{30} = 2.75\) | B1 | |
| \(e^{-2.75}(1 + 2.75 + \frac{2.75^2}{2})\) | M1 | Allow any \(\lambda\); allow one end error |
| \(= 0.481\) (3 sf) | A1 [3] | As final answer. Accept 0.482 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\lambda = 3.3 \times \frac{365}{30} = 40.15\) | B1 | Accept 40.1 or 40.2 |
| \((X \sim Po(40.15) \Rightarrow X \sim N(40.15,\ 40.15))\); \(\frac{50.5 - \text{"40.15"}}{\sqrt{\text{"40.15"}}}\) \((= 1.633)\) | M1 | Allow with incorrect or no cc OR no \(\sqrt{\phantom{x}}\) sign |
| \(1 - \Phi(\text{"1.633"}) = 0.0513\) (3 sf) | M1 A1 [4] | For correct area consistent with their working; accept 0.0512 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\lambda > 15\) | B1 [1] | Or similar |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\lambda = \frac{73}{30}\) oe or \(1.1 + 1.33 = 2.43\) (3 sf) | B1 | |
| \(1 - e^{-2.43}\left(1 + 2.43 + \frac{2.43^2}{2} + \frac{2.43^3}{3!}\right)\) | M1 | Allow any \(\lambda\); allow one end error |
| \(= 0.228\) (3 sf) | A1 [3] |
## Question 6:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\lambda = 3.3 \times \frac{25}{30} = 2.75$ | **B1** | |
| $e^{-2.75}(1 + 2.75 + \frac{2.75^2}{2})$ | **M1** | Allow any $\lambda$; allow one end error |
| $= 0.481$ (3 sf) | **A1** [3] | As final answer. Accept 0.482 |
### Part (ii)(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\lambda = 3.3 \times \frac{365}{30} = 40.15$ | **B1** | Accept 40.1 or 40.2 |
| $(X \sim Po(40.15) \Rightarrow X \sim N(40.15,\ 40.15))$; $\frac{50.5 - \text{"40.15"}}{\sqrt{\text{"40.15"}}}$ $(= 1.633)$ | **M1** | Allow with incorrect or no cc OR no $\sqrt{\phantom{x}}$ sign |
| $1 - \Phi(\text{"1.633"}) = 0.0513$ (3 sf) | **M1 A1** [4] | For correct area consistent with their working; accept 0.0512 |
### Part (ii)(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\lambda > 15$ | **B1** [1] | Or similar |
### Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\lambda = \frac{73}{30}$ oe or $1.1 + 1.33 = 2.43$ (3 sf) | **B1** | |
| $1 - e^{-2.43}\left(1 + 2.43 + \frac{2.43^2}{2} + \frac{2.43^3}{3!}\right)$ | **M1** | Allow any $\lambda$; allow one end error |
| $= 0.228$ (3 sf) | **A1** [3] | |
---6 The battery in Sue's phone runs out at random moments. Over a long period, she has found that the battery runs out, on average, 3.3 times in a 30-day period.
\begin{enumerate}[label=(\roman*)]
\item Find the probability that the battery runs out fewer than 3 times in a 25-day period.
\item (a) Use an approximating distribution to find the probability that the battery runs out more than 50 times in a year ( 365 days).\\
(b) Justify the approximating distribution used in part (ii)(a).
\item Independently of her phone battery, Sue's computer battery also runs out at random moments. On average, it runs out twice in a 15-day period. Find the probability that the total number of times that her phone battery and her computer battery run out in a 10-day period is at least 4 .
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2016 Q6 [11]}}