| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2014 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Distribution |
| Type | State distribution and mean |
| Difficulty | Moderate -0.5 This is a straightforward application of standard exponential distribution formulas. Part (i) requires recognizing λ=0.01 so E(T)=100, part (ii) uses the standard median formula for exponential distributions, and the final part is a direct probability calculation P(T>20) using the exponential CDF. While it's a Further Maths topic, the question requires only recall of standard results and routine calculation with no problem-solving insight needed. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(E(T) = \frac{1}{0.01} = 100\) | B1 | State or find \(E(T)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\left[-e^{-0.01t}\right]_0^m = \frac{1}{2}\) | M1 | State or use equation for median \(m\) of \(T\); A.E.F. |
| \(e^{-0.01m} = \frac{1}{2}\), \(m = 100\ln 2 = 69.3\) | M1 A1 | Find value of \(m\) |
| \(P(T > 20) = 1-(1-e^{-0.2}) = e^{-0.2}\) or \(0.819\) | M1 A1 | Find \(P(T>20)\); S.R. B1 for \(0.181\) |
# Question 7:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $E(T) = \frac{1}{0.01} = 100$ | B1 | State or find $E(T)$ |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\left[-e^{-0.01t}\right]_0^m = \frac{1}{2}$ | M1 | State or use equation for median $m$ of $T$; A.E.F. |
| $e^{-0.01m} = \frac{1}{2}$, $m = 100\ln 2 = 69.3$ | M1 A1 | Find value of $m$ |
| $P(T > 20) = 1-(1-e^{-0.2}) = e^{-0.2}$ or $0.819$ | M1 A1 | Find $P(T>20)$; S.R. B1 for $0.181$ |
---7 The time, $T$ seconds, between successive cars passing a particular checkpoint on a wide road has probability density function f given by
$$\mathrm { f } ( t ) = \begin{cases} \frac { 1 } { 100 } \mathrm { e } ^ { - 0.01 t } & t \geqslant 0 \\ 0 & \text { otherwise } . \end{cases}$$
(i) State the expected value of $T$.\\
(ii) Find the median value of $T$.
Sally wishes to cross the road at this checkpoint and she needs 20 seconds to complete the crossing. She decides to start out immediately after a car passes. Find the probability that she will complete the crossing before the next car passes.
\hfill \mbox{\textit{CAIE FP2 2014 Q7 [6]}}