| Exam Board | OCR |
|---|---|
| Module | FS1 AS (Further Statistics 1 AS) |
| Year | 2018 |
| Session | March |
| Marks | 6 |
| Topic | Continuous Uniform Random Variables |
| Type | Find parameters from given statistics |
| Difficulty | Standard +0.3 This question tests standard properties of Poisson and uniform distributions with straightforward calculations. Part (i)(a) requires recalling that SD = √mean for Poisson; part (i)(b) involves scaling the parameter and using tables; part (ii) requires finding N from E(W) = (1+N)/2 = 6.5, then calculating a simple probability. All steps are routine applications of formulas with no problem-solving insight required, making it slightly easier than average. |
| Spec | 5.02e Discrete uniform distribution5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| \(\sqrt{6.5}\) or \(2.5495\ldots\) | B1 | Exact or awrt 2.55 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\text{Po}(26)\) | M1 | Stated or implied |
| For using Po(26) and 29 or 30 | M1 | soi |
| \(P(< 30) = 0.759(26\ldots)\) | A1 | Awrt 0.759 |
| Answer | Marks | Guidance |
|---|---|---|
| \(N = 12\) | M1 | 2.2a |
| \(P(> 3) = \frac{9}{12} = \frac{3}{4}\) or \(0.75\) | A1 | 1.1 |
### Part (i)(a)
$\sqrt{6.5}$ or $2.5495\ldots$ | **B1** | Exact or awrt 2.55
### Part (i)(b)
$\text{Po}(26)$ | **M1** | Stated or implied
For using Po(26) and 29 or 30 | **M1** | soi
$P(< 30) = 0.759(26\ldots)$ | **A1** | Awrt 0.759
### Part (ii)
$N = 12$ | **M1** | 2.2a
$P(> 3) = \frac{9}{12} = \frac{3}{4}$ or $0.75$ | **A1** | 1.1 | SC: $N = 13$, $P(> 3) = \frac{9}{13}$; B1
---2 The number of calls received by a customer service department in 30 minutes is denoted by $W$. It is known that $\mathrm { E } ( W ) = 6.5$.\\
(i) It is given that $W$ has a Poisson distribution.
\begin{enumerate}[label=(\alph*)]
\item Write down the standard deviation of $W$.
\item Find the probability that the total number of calls received in a randomly chosen period of 2 hours is less than 30 .\\
(ii) It is given instead that $W$ has a uniform distribution on $[ 1 , N ]$. Calculate the value of $\mathrm { P } ( W > 3 )$.
\end{enumerate}
\hfill \mbox{\textit{OCR FS1 AS 2018 Q2 [6]}}