| 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 | Calculate probability with given parameter |
| Difficulty | Standard +0.3 This is a straightforward application of standard exponential distribution properties. Part (i) requires recognizing λ=0.01 so E(T)=100 (direct recall). Part (ii) uses the standard median formula for exponential distributions. The final part requires computing P(T>20) using the exponential CDF, which is a routine calculation. All parts are textbook-standard with no novel insight required, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks |
|---|---|
| (ii) | 1 |
| Answer | Marks |
|---|---|
| S.R. B1 for 0⋅181 = e –0.2 or 0⋅819 M1 A1 | 1 |
| Answer | Marks |
|---|---|
| 2 | [6] |
Question 7:
--- 7 (i)
(ii) ---
7 (i)
(ii) | 1
State or find E(T): E(T) = = 100 B1
0.01
[ −e−0.01t ] m 1
State or use eqn. for median m of T: = (A.E.F.) M1
0
2
1
Find value of m: e -0.01m = , m = 100ln 2 = 69⋅3 M1 A1
2
–0.2
Find P(T > 20): P(T > 20) = 1 – (1 – e )
S.R. B1 for 0⋅181 = e –0.2 or 0⋅819 M1 A1 | 1
3
2 | [6]The time, $T$ seconds, between successive cars passing a particular checkpoint on a wide road has probability density function f given by
$$f(t) = \begin{cases} \frac{1}{100}e^{-0.01t} & t \geq 0, \\ 0 & \text{otherwise.} \end{cases}$$
\begin{enumerate}[label=(\roman*)]
\item State the expected value of $T$. [1]
\item Find the median value of $T$. [3]
\end{enumerate}
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. [2]
\hfill \mbox{\textit{CAIE FP2 2014 Q7 [6]}}