| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2016 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating the Binomial to the Poisson distribution |
| Type | Find parameter from Poisson probabilities |
| Difficulty | Standard +0.3 This is a straightforward multi-part question testing standard probability distribution manipulations. Part (a) requires writing binomial probabilities and solving a simple equation; part (b) involves Poisson probability ratios leading to a quadratic; part (c) is direct application of normal approximation conditions (np=32, finding variance). All parts are routine textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04d Normal approximation to binomial5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance5.02d Binomial: mean np and variance np(1-p)5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.02l Poisson conditions: for modelling5.02m Poisson: mean = variance = lambda |
| Answer | Marks | Guidance |
|---|---|---|
| \(120(p)^3(1-p)^7\) | B1 | Allow equivalent expressions e.g. \(10C3(p)^3(1-p)^{10-3}\) |
| Answer | Marks | Guidance |
|---|---|---|
| \([10C3](p)^3(1-p)^7 = [10C7]16(p)^7(1-p)^3\) or their (a)(i) \(=[10C7]16(p)^7(1-p)^3\) | M1 | Correct equation ft their (a)(i), condone missing binomial coefficients but 16 must be on correct side. Condone numerical slips. |
| \((1-p)^4 = 16(p)^4 \Rightarrow (1-p) = 2(p)\) | M1 | Attempt to solve as linear equation in \(p\). Must deal with algebraic terms correctly. |
| \(p = \frac{1}{3}\) | A1 | For \(\frac{1}{3}\) or exact equivalent. Allow 3/3 for correct answer only in (ii) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{e^{-\lambda}\lambda^3}{3!} = 5\frac{e^{-\lambda}\lambda^5}{5!}\) | M1 | Correct equation |
| \(4 = \lambda^2\) | M1 | Attempt to solve as far as \(\lambda^2 = k\) or \(\lambda = \sqrt{k}\). Allow numerical slips. |
| \(\lambda = 2\) | A1 | For \(\lambda = 2\) only |
| Answer | Marks | Guidance |
|---|---|---|
| \(np = 32\) | M1 | Use of \(np = 32\). Allow any value of \(p\) provided \(0 < p < 1\) |
| \(n = 80\) | A1 | |
| \(\alpha = 19.2\) | A1 |
## Question 2:
### Part (a)(i):
$120(p)^3(1-p)^7$ | B1 | Allow equivalent expressions e.g. $10C3(p)^3(1-p)^{10-3}$
### Part (a)(ii):
$[10C3](p)^3(1-p)^7 = [10C7]16(p)^7(1-p)^3$ or their (a)(i) $=[10C7]16(p)^7(1-p)^3$ | M1 | Correct equation ft their (a)(i), condone missing binomial coefficients but 16 must be on correct side. Condone numerical slips.
$(1-p)^4 = 16(p)^4 \Rightarrow (1-p) = 2(p)$ | M1 | Attempt to solve as linear equation in $p$. Must deal with algebraic terms correctly.
$p = \frac{1}{3}$ | A1 | For $\frac{1}{3}$ or exact equivalent. Allow 3/3 for correct answer only in (ii)
**NB1:** If 16 on wrong side get $p = \frac{2}{3}$, score M0M1A0
**NB2:** If no 16 or 16 disappears and they get $p = 0.5$, score $2^{nd}$ M1 A0
### Part (b):
$\frac{e^{-\lambda}\lambda^3}{3!} = 5\frac{e^{-\lambda}\lambda^5}{5!}$ | M1 | Correct equation
$4 = \lambda^2$ | M1 | Attempt to solve as far as $\lambda^2 = k$ or $\lambda = \sqrt{k}$. Allow numerical slips.
$\lambda = 2$ | A1 | For $\lambda = 2$ only
**NB1:** If 5 on wrong side get $\lambda = 10$, score M0M1A0
**NB2:** If no 5 or 5 disappears, get $\lambda^2 = 20$ or $\lambda = \sqrt{20} = 2\sqrt{5}$, score $2^{nd}$ M1 A0
### Part (c):
$np = 32$ | M1 | Use of $np = 32$. Allow any value of $p$ provided $0 < p < 1$
$n = 80$ | A1 |
$\alpha = 19.2$ | A1 |
---2. The random variable $X \sim \mathrm {~B} ( 10 , p )$
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Write down an expression for $\mathrm { P } ( X = 3 )$ in terms of $p$
\item Find the value of $p$ such that $\mathrm { P } ( X = 3 )$ is 16 times the value of $\mathrm { P } ( X = 7 )$
The random variable $Y \sim \operatorname { Po } ( \lambda )$
\end{enumerate}\item Find the value of $\lambda$ such that $\mathrm { P } ( Y = 3 )$ is 5 times the value of $\mathrm { P } ( Y = 5 )$
The random variable $W \sim \mathrm {~B} ( n , 0.4 )$
\item Find the value of $n$ and the value of $\alpha$ such that $W$ can be approximated by the normal distribution, $\mathrm { N } ( 32 , \alpha )$\\
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2016 Q2 [10]}}