| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2002 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Poisson approximation justification or comparison |
| Difficulty | Standard +0.3 This is a straightforward application of the Poisson distribution with standard bookwork justification in part (a), routine probability calculations in (b), scaling the parameter in (c), and a standard normal approximation in (d). All techniques are textbook exercises requiring no novel insight, though the multi-part structure and normal approximation push it slightly above average difficulty. |
| Spec | 2.04d Normal approximation to binomial5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Failed connections occur singly, independently and at a constant rate of 3 per hour, randomly | B1, B1 | Any two conditions |
| Answer | Marks |
|---|---|
| \(X\) is number of failed connections per hour. \(P(X=0) = 0.0498\) | M1A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X > 4) = 1 - 0.8153 = 0.1847\) | M1A1 | Require '1 minus', 0.1847 |
| Answer | Marks |
|---|---|
| \(X \sim Po(24)\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(Y\) is number of users that fail to connect at first attempt, \(Y \sim N(24, 24)\) | B1, B1 | Normal, both parameters |
| \(P(Y \geq 12) = 1 - P\!\left(Z < \frac{11.5 - 24}{\sqrt{24}}\right)\) | M1, A1 | From above, all correct |
| \(= P(Z < -2.55)\) | A1 | \(-2.55\) |
| \(= 0.9946\) | A1 |
# Question 5:
## Part (a)
| Failed connections occur singly, independently and at a constant rate of 3 per hour, randomly | B1, B1 | Any two conditions |
## Part (b)(i)
| $X$ is number of failed connections per hour. $P(X=0) = 0.0498$ | M1A1 | |
## Part (b)(ii)
| $P(X > 4) = 1 - 0.8153 = 0.1847$ | M1A1 | Require '1 minus', 0.1847 |
## Part (c)
| $X \sim Po(24)$ | B1 | |
## Part (d)
| $Y$ is number of users that fail to connect at first attempt, $Y \sim N(24, 24)$ | B1, B1 | Normal, both parameters |
| $P(Y \geq 12) = 1 - P\!\left(Z < \frac{11.5 - 24}{\sqrt{24}}\right)$ | M1, A1 | From above, all correct |
| $= P(Z < -2.55)$ | A1 | $-2.55$ |
| $= 0.9946$ | A1 | |
---5. An Internet service provider has a large number of users regularly connecting to its computers. On average only 3 users every hour fail to connect to the Internet at their first attempt.
\begin{enumerate}[label=(\alph*)]
\item Give 2 reasons why a Poisson distribution might be a suitable model for the number of failed connections every hour.
\item Find the probability that in a randomly chosen hour
\begin{enumerate}[label=(\roman*)]
\item all Internet users connect at their first attempt,
\item more than 4 users fail to connect at their first attempt.
\end{enumerate}\item Write down the distribution of the number of users failing to connect at their first attempt in an 8-hour period.
\item Using a suitable approximation, find the probability that 12 or more users fail to connect at their first attempt in a randomly chosen 8-hour period.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2002 Q5 [13]}}