| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2018 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Distribution |
| Type | State distribution and mean |
| Difficulty | Standard +0.3 This is a straightforward application of the exponential distribution with standard bookwork parts: writing down the pdf from the given mean, calculating a probability using the exponential cdf, and finding the median by solving a simple equation. All three parts require only direct recall and routine manipulation of the exponential distribution formula, making it slightly easier than average for A-level Further Maths. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf |
| Answer | Marks | Guidance |
|---|---|---|
| 7(i) | f(t) = (1/500) exp (– t/500) [0 otherwise] (AEF) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| 7(ii) | F(t) = 1 – exp (– t/500) | M1 |
| P(T > 750) = 1 – F(750) = exp (– 750/500) = 0.223 | M1A1 | Find P(T > 750) (M0 for F(750)) |
| Answer | Marks | Guidance |
|---|---|---|
| 7(iii) | 1 – exp (– m/500) = ½ or exp (– m/500) = ½ | M1 |
| m = 500 ln 2 = 347 [hours] | M1A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 7:
--- 7(i) ---
7(i) | f(t) = (1/500) exp (– t/500) [0 otherwise] (AEF) | B1 | State pdf of T for t ⩾ 0
1
--- 7(ii) ---
7(ii) | F(t) = 1 – exp (– t/500) | M1 | Find or imply F(t)
P(T > 750) = 1 – F(750) = exp (– 750/500) = 0.223 | M1A1 | Find P(T > 750) (M0 for F(750))
3
--- 7(iii) ---
7(iii) | 1 – exp (– m/500) = ½ or exp (– m/500) = ½ | M1 | Find median value m of T from F(m) = ½
m = 500 ln 2 = 347 [hours] | M1A1
3
Question | Answer | Marks | GuidanceThe random variable $T$ is the lifetime, in hours, of a particular type of battery. It is given that $T$ has a negative exponential distribution with mean 500 hours.
\begin{enumerate}[label=(\roman*)]
\item Write down the probability density function of $T$. [1]
\item Find the probability that a randomly chosen battery of this type has a lifetime of more than 750 hours. [3]
\item Find the median value of $T$. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE FP2 2018 Q7 [7]}}