| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2016 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Distribution |
| Type | Find threshold for given probability |
| Difficulty | Moderate -0.5 This is a straightforward application of exponential distribution properties requiring only standard recall and basic algebraic manipulation. Finding the parameter from the mean uses a standard formula (mean = 1/a), part (ii) is direct substitution into F(x), and part (iii) requires solving 1 - F(d) = 0.75 using logarithms. All steps are routine textbook exercises with no problem-solving insight required, making it slightly easier than average. |
| Spec | 1.06e Logarithm as inverse: ln(x) inverse of e^x5.03a Continuous random variables: pdf and cdf5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(a = 1/10\,000\) or \(10^{-4}\) | B1 | Find \(a\) from mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(1 - e^{-15000a} = 1 - e^{-1.5} = 0.777\) | M1 A1 | Find \(P(X < 15\,000)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(1 - (1 - e^{-ad}) = 0.75\) | M1 | Formulate condition for \(d\); (M0 for \(1 - e^{-ad} = 0.75\), giving \(d = 13\,900\)) |
| \(d = -(\ln 0.75)/a = 2877\) or \(2880\) | A1 A1 | Rearrange and take logs to give \(d\) |
# Question 5:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $a = 1/10\,000$ or $10^{-4}$ | B1 | Find $a$ from mean |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $1 - e^{-15000a} = 1 - e^{-1.5} = 0.777$ | M1 A1 | Find $P(X < 15\,000)$ |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $1 - (1 - e^{-ad}) = 0.75$ | M1 | Formulate condition for $d$; (M0 for $1 - e^{-ad} = 0.75$, giving $d = 13\,900$) |
| $d = -(\ln 0.75)/a = 2877$ or $2880$ | A1 A1 | Rearrange and take logs to give $d$ |5 The distance, $X \mathrm {~km}$, completed by a new car before any mechanical fault occurs has distribution function F given by
$$\mathrm { F } ( x ) = \begin{cases} 1 - \mathrm { e } ^ { - a x } & x \geqslant 0 \\ 0 & \text { otherwise } \end{cases}$$
where $a$ is a positive constant. The mean value of $X$ is 10000 . Find\\
(i) the value of $a$,\\
(ii) the probability that a new car completes less than 15000 km before any mechanical fault occurs.
The probability that a new car completes at least $d \mathrm {~km}$ before any mechanical fault occurs is 0.75 .\\
(iii) Find the value of $d$.
\hfill \mbox{\textit{CAIE FP2 2016 Q5 [6]}}