| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2014 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Distribution |
| Type | Find threshold for given probability |
| Difficulty | Standard +0.3 This question involves standard exponential distribution calculations: writing the pdf from the mean, computing P(T > 2000) using the exponential survival function, and finding a threshold by solving (e^(-t/1000))^10 ≥ 0.9. While it requires understanding of independence and taking logarithms, these are routine techniques for Further Maths students with no novel problem-solving required. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf |
| Answer | Marks | Guidance |
|---|---|---|
| \(f(t) = 0.001\exp(-0.001t)\ \ (t \geq 0)\) | B1 | State probability density function of \(T\) |
| \([= 0 \quad \text{(otherwise or } t < 0)]\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(t > 2000) = 1 - F(2000)\) | M1 | Find \(P(T > 2000)\) |
| \(= 1 - (1 - e^{-2}) = e^{-2}\) or \(0.135\) | M1 A1 | S.R. \(1 - e^{-2} = 0.865\) earns B1 only (max 1/3) |
| \((\exp(-0.001t))^{10} \geq [or >]\ 0.9\) | M1 A1 | State inequality for \(t\) (lose A1 if \(=\) or \(\leq\)) |
| \(t_{\max} = (\ln 0.9)/(-0.01) = 10.5\) | M1 A1 | Solve for \(t_{\max}\); omitting power 10 earns 0/4; using \(1-(\exp(-0.001t))^{10}\) can earn M1 A0 M1 A0 only |
# Question 7:
**Part (i):**
| $f(t) = 0.001\exp(-0.001t)\ \ (t \geq 0)$ | B1 | State probability density function of $T$ |
| $[= 0 \quad \text{(otherwise or } t < 0)]$ | | |
**Part (ii):**
| $P(t > 2000) = 1 - F(2000)$ | M1 | Find $P(T > 2000)$ |
| $= 1 - (1 - e^{-2}) = e^{-2}$ or $0.135$ | M1 A1 | S.R. $1 - e^{-2} = 0.865$ earns B1 only (max 1/3) |
| $(\exp(-0.001t))^{10} \geq [or >]\ 0.9$ | M1 A1 | State inequality for $t$ (lose A1 if $=$ or $\leq$) |
| $t_{\max} = (\ln 0.9)/(-0.01) = 10.5$ | M1 A1 | Solve for $t_{\max}$; omitting power 10 earns 0/4; using $1-(\exp(-0.001t))^{10}$ can earn M1 A0 M1 A0 only |
---7 The random variable $T$ is the lifetime, in hours, of a randomly chosen decorative light bulb of a particular type. It is given that $T$ has a negative exponential distribution with mean 1000 hours.\\
(i) Write down the probability density function of $T$.\\
(ii) Find the probability that a randomly chosen bulb of this type has a lifetime of more than 2000 hours.
A display uses 10 randomly chosen bulbs of this type, and they are all switched on simultaneously. Find the greatest value of $t$ such that the probability that they are all alight at time $t$ hours is at least 0.9 .
\hfill \mbox{\textit{CAIE FP2 2014 Q7 [8]}}