| Exam Board | Edexcel |
|---|---|
| Module | FS1 AS (Further Statistics 1 AS) |
| Year | 2020 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sum of Poisson processes |
| Type | Conditional probability given total |
| Difficulty | Standard +0.3 This is a straightforward application of standard Poisson distribution properties: (a) requires stating that independent Poisson processes sum to Poisson with combined mean, (b) uses conditional probability with binomial approximation or direct calculation, and (c) is a routine hypothesis test. All parts follow textbook procedures with no novel insight required, though it's slightly above average difficulty due to being Further Maths content and requiring careful handling of the conditional probability in part (b). |
| Spec | 5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02l Poisson conditions: for modelling5.02m Poisson: mean = variance = lambda5.02n Sum of Poisson variables: is Poisson5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \([X \sim \text{Po}(8),\ Y \sim \text{Po}(3)]\); \([X+Y] \sim \text{Po}(11)\) | B1 | Correct model |
| The number of cyclists travelling eastbound is independent of the number of cyclists travelling westbound. | B1 | Correct modelling assumption in context (must mention cyclists) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{P(X=11)\times P(Y=1) + P(X=12)\times P(Y=0)}{P(X+Y=12)}\) | M1, M1 | M1: ratio with denominator \(P(X+Y=12)\); M1: correct numerator expression |
| \(= 0.1204\ldots\) awrt 0.120 | A1 | awrt 0.120, accept 0.12 with correct working. Alternative: \(C \sim B(12, \frac{8}{11})\); \(P(C \geq 11) = 1 - P(C \leq 10)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0\): \(\lambda = 11\) or \(\mu = 22\); \(H_1\): \(\lambda < 11\) or \(\mu < 22\) | B1 | Both hypotheses with \(\lambda\) or \(\mu\) |
| \((E+W) \sim \text{Po}(22)\); \(P(E+W \leq 14)\) [= awrt 0.048] | M1 | Using Po(22) to calculate \(P(E+W \leq 14)\) |
| (Reject \(H_0\).) There is evidence that the rate of cyclists has decreased. | A1 | Fully correct conclusion with awrt 0.048 or CR: \(E+W \leq 14\), drawing inference in context |
# Question 4:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $[X \sim \text{Po}(8),\ Y \sim \text{Po}(3)]$; $[X+Y] \sim \text{Po}(11)$ | B1 | Correct model |
| The number of cyclists travelling eastbound is independent of the number of cyclists travelling westbound. | B1 | Correct modelling assumption in context (must mention cyclists) |
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{P(X=11)\times P(Y=1) + P(X=12)\times P(Y=0)}{P(X+Y=12)}$ | M1, M1 | M1: ratio with denominator $P(X+Y=12)$; M1: correct numerator expression |
| $= 0.1204\ldots$ awrt **0.120** | A1 | awrt 0.120, accept 0.12 with correct working. Alternative: $C \sim B(12, \frac{8}{11})$; $P(C \geq 11) = 1 - P(C \leq 10)$ |
## Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0$: $\lambda = 11$ or $\mu = 22$; $H_1$: $\lambda < 11$ or $\mu < 22$ | B1 | Both hypotheses with $\lambda$ or $\mu$ |
| $(E+W) \sim \text{Po}(22)$; $P(E+W \leq 14)$ [= awrt 0.048] | M1 | Using Po(22) to calculate $P(E+W \leq 14)$ |
| (Reject $H_0$.) There is evidence that the **rate** of **cyclists** has decreased. | A1 | Fully correct conclusion with awrt 0.048 or CR: $E+W \leq 14$, drawing inference in context |
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\item During the morning, the number of cyclists passing a particular point on a cycle path in a 10-minute interval travelling eastbound can be modelled by a Poisson distribution with mean 8
\end{enumerate}
The number of cyclists passing the same point in a 10 -minute interval travelling westbound can be modelled by a Poisson distribution with mean 3\\
(a) Suggest a model for the total number of cyclists passing the point on the cycle path in a 10-minute interval, stating a necessary assumption.
Given that exactly 12 cyclists pass the point in a 10 -minute interval,\\
(b) find the probability that at least 11 are travelling eastbound.
After some roadworks were completed, the total number of cyclists passing the point in a randomly selected 20-minute interval one morning is found to be 14\\
(c) Test, at the $5 \%$ level of significance, whether there is evidence of a decrease in the rate of cyclists passing the point.\\
State your hypotheses clearly.
\hfill \mbox{\textit{Edexcel FS1 AS 2020 Q4 [8]}}