| Exam Board | Edexcel |
|---|---|
| Module | FS1 (Further Statistics 1) |
| Year | 2020 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating the Binomial to the Poisson distribution |
| Type | Justify Poisson approximation only |
| Difficulty | Moderate -0.5 This is a straightforward Further Statistics question testing standard conditions for Poisson approximation (n large, p small, np moderate) and basic properties of Poisson distributions. Part (a) requires recalling the rule that p should be small (here p=0.4 is too large), part (b) asks for independence assumption (standard bookwork), and part (c) involves routine calculation using tables. While it's Further Maths content, it requires only recall and direct application of learned criteria with minimal problem-solving, making it easier than average overall. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02d Binomial: mean np and variance np(1-p)5.02e Discrete uniform distribution5.02n Sum of Poisson variables: is Poisson |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Requires large \(n\)/small \(p\) so not a good approximation | B1 | Correct reason why the model would not be appropriate and correct conclusion. Condone e.g. '\(p\) is close to 0.5' for \(p\) is not small. Mean is not equal to variance on its own in B0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(X\) and \(Y\) must be independent | B1 | Correct explanation mentioning independence (oe). Ignore extraneous comments |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(P(X + Y < 2.4)\) from \(Po(7)\ [P(X + Y \leq 2)]\) | M1 | Using \(Po(7)\) with 2.4 |
| \(= 0.029636\ldots\) awrt \(\mathbf{0.0296}\) | A1 | awrt 0.0296 |
## Question 2:
### Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Requires large $n$/small $p$ so not a good approximation | B1 | Correct reason why the model would not be appropriate and correct conclusion. Condone e.g. '$p$ is close to 0.5' for $p$ is not small. Mean is not equal to variance on its own in B0 |
### Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $X$ and $Y$ must be independent | B1 | Correct explanation mentioning independence (oe). Ignore extraneous comments |
### Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $P(X + Y < 2.4)$ from $Po(7)\ [P(X + Y \leq 2)]$ | M1 | Using $Po(7)$ with 2.4 |
| $= 0.029636\ldots$ awrt $\mathbf{0.0296}$ | A1 | awrt 0.0296 |\begin{enumerate}
\item The discrete random variables $W , X$ and $Y$ are distributed as follows
\end{enumerate}
$$W \sim \mathrm {~B} ( 10,0.4 ) \quad X \sim \operatorname { Po } ( 4 ) \quad Y \sim \operatorname { Po } ( 3 )$$
(a) Explain whether or not $\mathrm { Po } ( 4 )$ would be a good approximation to $\mathrm { B } ( 10,0.4 )$\\
(b) State the assumption required for $X + Y$ to be distributed as $\operatorname { Po } ( 7 )$
Given the assumption in part (b) holds,\\
(c) find $\mathrm { P } ( X + Y < \operatorname { Var } ( W ) )$
\hfill \mbox{\textit{Edexcel FS1 2020 Q2 [4]}}