| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2011 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Distribution |
| Type | State distribution and mean |
| Difficulty | Moderate -0.8 This is a straightforward exponential distribution question requiring only recognition of the standard form (λe^(-λx)) to state the mean (1/λ = 100), then routine calculations for median using ln(2)/λ and a simple probability integral. All steps are direct applications of standard formulas with no problem-solving or insight required. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| State or find \(E(X)\): \(E(X) = 1/0{\cdot}01\ \text{or}\ 100\) | B1 | [Subtotal: 1] |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Integrate \(f(x)\) to find median \(m\): \(\int_0^m f(x)\,dx = 1 - e^{-0.01m} = \frac{1}{2}\) | M1 A1 | |
| Solve for \(m\): \(m = 100\ln 2\ \text{or}\ 69{\cdot}3\) | A1 | [Subtotal: 3] |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Integrate \(f(x)\) to find probability: \(\int_m^{100} f(x)\,dx = \frac{1}{2} - e^{-1} = 0{\cdot}132\) | M1 A1 | [Subtotal: 2] |
## Question 5:
### Part (i):
| Working/Answer | Mark | Guidance |
|---|---|---|
| State or find $E(X)$: $E(X) = 1/0{\cdot}01\ \text{or}\ 100$ | B1 | [Subtotal: 1] |
### Part (ii):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Integrate $f(x)$ to find median $m$: $\int_0^m f(x)\,dx = 1 - e^{-0.01m} = \frac{1}{2}$ | M1 A1 | |
| Solve for $m$: $m = 100\ln 2\ \text{or}\ 69{\cdot}3$ | A1 | [Subtotal: 3] |
### Part (iii):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Integrate $f(x)$ to find probability: $\int_m^{100} f(x)\,dx = \frac{1}{2} - e^{-1} = 0{\cdot}132$ | M1 A1 | [Subtotal: 2] |
**Total: 6**
---5 The continuous random variable $X$ has probability density function f given by
$$\mathrm { f } ( x ) = \begin{cases} 0.01 \mathrm { e } ^ { - 0.01 x } & x \geqslant 0 \\ 0 & x < 0 \end{cases}$$
(i) State the value of $\mathrm { E } ( X )$.\\
(ii) Find the median value of $X$.\\
(iii) Find the probability that $X$ lies between the median and the mean.
\hfill \mbox{\textit{CAIE FP2 2011 Q5 [6]}}