| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2012 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sum of Poisson processes |
| Type | Basic sum of two Poissons |
| Difficulty | Standard +0.3 This is a straightforward application of Poisson distribution with scaled parameters. Part (i) requires recognizing that independent Poisson variables sum to another Poisson (λ = 0.32×8 + 0.45×12 = 8.0) and computing P(X ≥ 3). Part (ii) is a direct calculation with λ = 0.32×155 = 49.6. Both parts are standard textbook exercises requiring only recall of properties and calculator work, making this slightly easier than average. |
| Spec | 5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.02n Sum of Poisson variables: is Poisson |
| Answer | Marks | Guidance |
|---|---|---|
| \(\lambda = 8 \times 0.32 + 12 \times 0.45\ (= 7.96)\) | M1 | |
| \(1 - e^{-7.96}(1 + 7.96 + \frac{7.96^2}{2})\) | M1 | \(1 - P(X \leq 2)\), any \(\lambda\) allow one end error |
| \(= 0.986\) (3 sfs) | A1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| \(\lambda = 155 \times 0.32 = 49.6\) | B1 | |
| \(N(49.6,\ 49.6)\) | M1 | \(N(\lambda,\ \lambda)\) any \(\lambda\). May be implied |
| \(\frac{34.5 - 49.6}{\sqrt{49.6}}\ (= -2.144)\) | M1 | Allow no or wrong cc & no \(\sqrt{}\) |
| \(\Phi(-2.144) = 1 - \Phi(2.144)\) | M1 | |
| \(= 0.016(0)\) | A1 [5] | Correct area consistent with their working |
## Question 4:
**(i)**
$\lambda = 8 \times 0.32 + 12 \times 0.45\ (= 7.96)$ | M1 |
$1 - e^{-7.96}(1 + 7.96 + \frac{7.96^2}{2})$ | M1 | $1 - P(X \leq 2)$, any $\lambda$ allow one end error
$= 0.986$ (3 sfs) | A1 [3] |
**(ii)**
$\lambda = 155 \times 0.32 = 49.6$ | B1 |
$N(49.6,\ 49.6)$ | M1 | $N(\lambda,\ \lambda)$ any $\lambda$. May be implied
$\frac{34.5 - 49.6}{\sqrt{49.6}}\ (= -2.144)$ | M1 | Allow no or wrong cc & no $\sqrt{}$
$\Phi(-2.144) = 1 - \Phi(2.144)$ | M1 |
$= 0.016(0)$ | A1 [5] | Correct area consistent with their working
---4 Bacteria of a certain type are randomly distributed in the water in two ponds, $A$ and $B$. The average numbers of bacteria per $\mathrm { cm } ^ { 3 }$ in $A$ and $B$ are 0.32 and 0.45 respectively.\\
(i) Samples of $8 \mathrm {~cm} ^ { 3 }$ of water from $A$ and $12 \mathrm {~cm} ^ { 3 }$ of water from $B$ are taken at random. Find the probability that the total number of bacteria in these samples is at least 3 .\\
(ii) Find the probability that in a random sample of $155 \mathrm {~cm} ^ { 3 }$ of water from $A$, the number of bacteria is less than 35 .
\hfill \mbox{\textit{CAIE S2 2012 Q4 [8]}}