| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2017 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Distribution |
| Type | Derive CDF from PDF |
| Difficulty | Moderate -0.8 This is a straightforward application of standard exponential distribution formulas. Part (i) requires integrating the PDF to get CDF (a routine calculus exercise), while parts (ii) and (iii) involve direct substitution into the CDF. The exponential distribution is a core topic with well-practiced techniques, making this easier than average despite being Further Maths content. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03e Find cdf: by integration5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| \(F(x) = \int f(x)\,dx = -e^{-0.2x} + c = 1 - e^{-0.2x}\) \((x \geq 0)\) | M1 | State, or integrate and use \(F(0)=0\) or \(F(x) \to 1\) as \(x \to \infty\) |
| and \(F(x) = 0\) \((x < 0\) or otherwise\()\) | A1 | to find \(F(x)\) (A0 if case \(x < 0\) omitted) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X > 2) = 1 - F(2) = e^{-0.4} = 0.670\) | M1 A1 | Find \(P(X > 2)\): (M0 for \(F(2)\)) |
| Answer | Marks | Guidance |
|---|---|---|
| \(1 - e^{-0.2m} = \frac{1}{2}\), \(e^{0.2m} = 2\) | M1 | Find median value \(m\) from \(F(m)\) or \(1 - F(m) = \frac{1}{2}\) |
| \(m = 5\ln 2\) or \(3.47\) | M1 A1 |
## Question 7(i):
$F(x) = \int f(x)\,dx = -e^{-0.2x} + c = 1 - e^{-0.2x}$ $(x \geq 0)$ | M1 | State, or integrate and use $F(0)=0$ or $F(x) \to 1$ as $x \to \infty$
and $F(x) = 0$ $(x < 0$ or otherwise$)$ | A1 | to find $F(x)$ (**A0** if case $x < 0$ omitted)
**Total: 2**
## Question 7(ii):
$P(X > 2) = 1 - F(2) = e^{-0.4} = 0.670$ | M1 A1 | Find $P(X > 2)$: (**M0** for $F(2)$)
**Total: 3**
## Question 7(iii):
$1 - e^{-0.2m} = \frac{1}{2}$, $e^{0.2m} = 2$ | M1 | Find median value $m$ from $F(m)$ or $1 - F(m) = \frac{1}{2}$
$m = 5\ln 2$ or $3.47$ | M1 A1 |
**Total: 3**
---7 The random variable $X$ has probability density function f given by
$$\mathrm { f } ( x ) = \begin{cases} 0.2 \mathrm { e } ^ { - 0.2 x } & x \geqslant 0 \\ 0 & \text { otherwise } \end{cases}$$
(i) Find the distribution function of $X$.\\
(ii) Find $\mathrm { P } ( X > 2 )$.\\
(iii) Find the median of $X$.\\
\hfill \mbox{\textit{CAIE FP2 2017 Q7 [7]}}