| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2018 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Cumulative distribution functions |
| Type | Interquartile range calculation |
| Difficulty | Moderate -0.3 This is a straightforward application of CDF properties requiring students to evaluate 1-F(2) and solve F(x)=0.25 and F(x)=0.75 using logarithms. While it involves exponentials and logs, these are routine manipulations with no conceptual difficulty beyond knowing the definition of quartiles and how to use a CDF, making it slightly easier than average. |
| Spec | 5.03e Find cdf: by integration5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X > 2) = 1 - F(2) = \exp(-0.8) = 0.449\) | M1 A1 | Find \(P(X > 2)\). M0 for \(F(2)\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(F(Q) = 1 - \exp(-0.4x) = \frac{1}{4}\) or \(\frac{3}{4}\) | M1 | Formulate equation for either quartile value \(Q\) |
| \(Q_1 = (\ln 4/3)/(0.4) \quad [= 0.7192]\) | A1 (AEF) | Find one [lower] quartile \(Q_1\) |
| \(Q_3 = (\ln 4)/(0.4) \quad [= 3.466]\) | A1 (AEF) | Find other [upper] quartile \(Q_3\) |
| \(Q_3 - Q_1 [= (\ln 3)/(0.4)] = 2.75\) | \(\text{A1}\sqrt{}\) | Find interquartile range (FT on \(Q_1\), \(Q_3\); allow \(Q_1 - Q_3\)) |
## Question 6(i):
| $P(X > 2) = 1 - F(2) = \exp(-0.8) = 0.449$ | M1 A1 | Find $P(X > 2)$. M0 for $F(2)$ |
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## Question 6(ii):
| $F(Q) = 1 - \exp(-0.4x) = \frac{1}{4}$ or $\frac{3}{4}$ | M1 | Formulate equation for either quartile value $Q$ |
|---|---|---|
| $Q_1 = (\ln 4/3)/(0.4) \quad [= 0.7192]$ | A1 (AEF) | Find one [lower] quartile $Q_1$ |
| $Q_3 = (\ln 4)/(0.4) \quad [= 3.466]$ | A1 (AEF) | Find other [upper] quartile $Q_3$ |
| $Q_3 - Q_1 [= (\ln 3)/(0.4)] = 2.75$ | $\text{A1}\sqrt{}$ | Find interquartile range (FT on $Q_1$, $Q_3$; allow $Q_1 - Q_3$) |
---6 The continuous random variable $X$ has distribution function given by
$$\mathrm { F } ( x ) = \begin{cases} 1 - \mathrm { e } ^ { - 0.4 x } & x \geqslant 0 \\ 0 & \text { otherwise } \end{cases}$$
(i) Find $\mathrm { P } ( X > 2 )$.\\
(ii) Find the interquartile range of $X$.\\
\hfill \mbox{\textit{CAIE FP2 2018 Q6 [6]}}