| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2005 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Single period normal approximation - scaled period (normal approximation only) |
| Difficulty | Easy -1.2 Part (a) requires simple recall of Poisson conditions. Parts (b)-(d) involve standard Poisson calculations and a routine normal approximation—all textbook exercises with no problem-solving insight required. This is easier than average for A-level. |
| Spec | 5.02i Poisson distribution: random events model5.02k Calculate Poisson probabilities5.02l Poisson conditions: for modelling5.02m Poisson: mean = variance = lambda5.02n Sum of Poisson variables: is Poisson5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| Misprints are random/independent, occur singly in space and at a constant rate | B1, B1 | Context, any 2 |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X=0) = e^{-2.5} = 0.08208\ldots = 0.0821\) | M1 | Po(2.5) |
| A1 | 0.0821 |
| Answer | Marks | Guidance |
|---|---|---|
| \(Y \sim \text{Po}(5)\) for 2 pages | B1 | Implied |
| \(P(Y > 7) = 1 - P(X \le 7)\) | M1 | Use of \(1 -\) and correct inequality |
| \(= 1 - 0.8666 = 0.1334\) | A1 | 0.1334 |
| Answer | Marks | Guidance |
|---|---|---|
| For 20 pages, \(Y \sim P_{\mu}(50)\) | B1 | \(P_{\mu}(50)\) |
| \(Y \sim N(50, 50)\) approx | B1 | \(N(50, 50)\) |
| \(P(Y < 40) = P(Y \le 39.5) = P\left(Z \le \frac{39.5-50}{\sqrt{50}}\right)\) | M1, M1 | cc \(\pm 0.5\), standardise above, all correct |
| \(= P(Z \le -1.4849) = 1 - 0.93 = 0.07\) | A1, A1 | awrt \(-1.48\), 0.07 |
**3(a)**
| Misprints are random/independent, occur singly in space and at a constant rate | B1, B1 | Context, any 2 |
**3(b)**
| $P(X=0) = e^{-2.5} = 0.08208\ldots = 0.0821$ | M1 | Po(2.5) |
| | A1 | 0.0821 |
**3(c)**
| $Y \sim \text{Po}(5)$ for 2 pages | B1 | Implied |
| $P(Y > 7) = 1 - P(X \le 7)$ | M1 | Use of $1 -$ and correct inequality |
| $= 1 - 0.8666 = 0.1334$ | A1 | 0.1334 |
**3(d)**
| For 20 pages, $Y \sim P_{\mu}(50)$ | B1 | $P_{\mu}(50)$ |
| $Y \sim N(50, 50)$ approx | B1 | $N(50, 50)$ |
| $P(Y < 40) = P(Y \le 39.5) = P\left(Z \le \frac{39.5-50}{\sqrt{50}}\right)$ | M1, M1 | cc $\pm 0.5$, standardise above, all correct |
| $= P(Z \le -1.4849) = 1 - 0.93 = 0.07$ | A1, A1 | awrt $-1.48$, 0.07 |
---3. The random variable $X$ is the number of misprints per page in the first draft of a novel.
\begin{enumerate}[label=(\alph*)]
\item State two conditions under which a Poisson distribution is a suitable model for $X$.
The number of misprints per page has a Poisson distribution with mean 2.5. Find the probability that
\item a randomly chosen page has no misprints,
\item the total number of misprints on 2 randomly chosen pages is more than 7 .
The first chapter contains 20 pages.
\item Using a suitable approximation find, to 2 decimal places, the probability that the chapter will contain less than 40 misprints.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2005 Q3 [14]}}