| Exam Board | OCR |
|---|---|
| Module | Further Statistics (Further Statistics) |
| Year | 2018 |
| Session | March |
| Marks | 6 |
| Topic | Poisson distribution |
| Type | Expectation and variance of Poisson-related expressions |
| Difficulty | Standard +0.3 This is a straightforward application of standard Poisson distribution properties: sum of independent Poissons, normal approximation for probability calculation, and variance of linear combinations. While it's Further Maths content, the techniques are routine and require no novel insight—just recall of formulas and careful arithmetic. |
| Spec | 5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.02n Sum of Poisson variables: is Poisson5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| Answer | Marks | Guidance |
|---|---|---|
| \(X + Y \sim \text{Po}(31)\) | M1 | Stated or implied |
| \(P(X + Y > 40) = 1 - P(X + Y \leq 40)\) | M1 | Allow M1A0 for 0.0678 |
| \(= 0.048728\) | A1 | Awrt 0.0487 |
| Answer | Marks | Guidance |
|---|---|---|
| \(4 \times 17 + 14\) | M1 | Allow M1 for \(2 \times 17 + 14\) |
| \(= 82\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(X\) and \(Y\) are independent | B1 | Doesn't need to be contextualised |
## Part (i)(a)
$X + Y \sim \text{Po}(31)$ | M1 | Stated or implied
$P(X + Y > 40) = 1 - P(X + Y \leq 40)$ | M1 | Allow M1A0 for 0.0678
$= 0.048728$ | A1 | Awrt 0.0487 | BC
## Part (i)(b)
$4 \times 17 + 14$ | M1 | Allow M1 for $2 \times 17 + 14$
$= 82$ | A1 |
## Part (ii)
$X$ and $Y$ are independent | B1 | Doesn't need to be contextualised1 The numbers of customers arriving at a ticket desk between 8 a.m. and 9 a.m. on a Monday morning and on a Tuesday morning are denoted by $X$ and $Y$ respectively. It is given that $X \sim \operatorname { Po } ( 17 )$ and $Y \sim \operatorname { Po } ( 14 )$.\\
(i) Find
\begin{enumerate}[label=(\alph*)]
\item $\mathrm { P } ( X + Y ) > 40$,
\item $\operatorname { Var } ( 2 X - Y )$.\\
(ii) State a necessary assumption for your calculations in part (i) to be valid.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics 2018 Q1 [6]}}