| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2012 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Distribution |
| Type | Find parameter from given information |
| Difficulty | Standard +0.3 This is a straightforward application of exponential distribution properties. Part (i) uses the standard formula σ = 1/λ, part (ii) is direct probability calculation using the given CDF, and part (iii) applies the median definition F(m) = 0.5. All parts require recall of standard results and routine substitution with no problem-solving insight needed. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\lambda = 1/(\text{standard deviation}) = 1/8\) | M1 A1 | State or find \(\lambda\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(e^{-5\lambda} - e^{-10\lambda} = e^{-0.625} - e^{-1.25} = 0.249\) | M1 A1 | Find \(F(10) - F(5)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(1 - e^{-\lambda m} = \frac{1}{2}\) | M1 | Formulate equation for median \(m\) |
| \(m = -8\ln\frac{1}{2} = 5.55\) (or \(5.54\)) | M1 A1 | Find value of \(m\) |
## Question 7:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\lambda = 1/(\text{standard deviation}) = 1/8$ | M1 A1 | State or find $\lambda$ |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $e^{-5\lambda} - e^{-10\lambda} = e^{-0.625} - e^{-1.25} = 0.249$ | M1 A1 | Find $F(10) - F(5)$ |
### Part (iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $1 - e^{-\lambda m} = \frac{1}{2}$ | M1 | Formulate equation for median $m$ |
| $m = -8\ln\frac{1}{2} = 5.55$ (or $5.54$) | M1 A1 | Find value of $m$ |
**Total: 7 marks**
---7 The waiting time, $T$ minutes, before a customer is served in a restaurant has distribution function F given by
$$\mathrm { F } ( t ) = \begin{cases} 1 - \mathrm { e } ^ { - \lambda t } & t \geqslant 0 \\ 0 & t < 0 \end{cases}$$
where $\lambda$ is a positive constant. The standard deviation of $T$ is 8 . Find\\
(i) the value of $\lambda$,\\
(ii) the probability that a customer has to wait between 5 and 10 minutes before being served,\\
(iii) the median value of $T$.
\hfill \mbox{\textit{CAIE FP2 2012 Q7 [7]}}