| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2019 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Cumulative distribution functions |
| Type | PDF to CDF derivation |
| Difficulty | Standard +0.3 This is a straightforward PDF to CDF question requiring integration of a simple polynomial function, followed by finding quartiles by solving F(x) = 0.25 and 0.75. While it involves multiple steps and some algebraic manipulation, the techniques are standard for Further Maths statistics with no conceptual challenges or novel problem-solving required. |
| Spec | 5.03e Find cdf: by integration5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(F(x) = \int f(x)\,dx = -\frac{3}{4x} + \frac{x}{4}\ [+c]\) | M1 | Find or state distribution function \(F(x)\) for \(1 \leq x \leq 3\) |
| \(= -\frac{3}{4x} + \frac{x}{4} + \frac{1}{2}\) or \(\frac{1}{4}(-\frac{3}{x} + x + 2)\) | A1 | Using \(F(1)=0\) or \(F(3)=1\) to find \(c\) if necessary |
| \(F(x) = 0\ (x < \text{or} \leq 1)\), \(F(x) = 1\ (x > \text{or} \geq 3)\) | A1 | State \(F(x)\) for other values of \(x\) |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\int_1^Q f(x)\,dx = -\frac{3}{4Q} + \frac{Q}{4} + \frac{1}{2} = \frac{1}{4}\) [or \(\frac{3}{4}\)] (AEF) | M1 | Formulate equation for either quartile value \(Q\) |
| \(Q^2 + [\text{or} -]\ Q - 3 = 0\) | A1 | |
| \(Q_1 = \frac{1}{2}(-1+\sqrt{13})\), \(Q_3 = \frac{1}{2}(1+\sqrt{13})\) | A1 A1 | Find lower quartile \(Q_1\) and upper quartile \(Q_3\) |
| \(Q_3 - Q_1 = 1\) | A1 | Find interquartile range |
| 5 |
## Question 7(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $F(x) = \int f(x)\,dx = -\frac{3}{4x} + \frac{x}{4}\ [+c]$ | M1 | Find or state distribution function $F(x)$ for $1 \leq x \leq 3$ |
| $= -\frac{3}{4x} + \frac{x}{4} + \frac{1}{2}$ or $\frac{1}{4}(-\frac{3}{x} + x + 2)$ | A1 | Using $F(1)=0$ or $F(3)=1$ to find $c$ if necessary |
| $F(x) = 0\ (x < \text{or} \leq 1)$, $F(x) = 1\ (x > \text{or} \geq 3)$ | A1 | State $F(x)$ for other values of $x$ |
| | **3** | |
## Question 7(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\int_1^Q f(x)\,dx = -\frac{3}{4Q} + \frac{Q}{4} + \frac{1}{2} = \frac{1}{4}$ [or $\frac{3}{4}$] (AEF) | M1 | Formulate equation for either quartile value $Q$ |
| $Q^2 + [\text{or} -]\ Q - 3 = 0$ | A1 | |
| $Q_1 = \frac{1}{2}(-1+\sqrt{13})$, $Q_3 = \frac{1}{2}(1+\sqrt{13})$ | A1 A1 | Find lower quartile $Q_1$ and upper quartile $Q_3$ |
| $Q_3 - Q_1 = 1$ | A1 | Find interquartile range |
| | **5** | |7 The continuous random variable $X$ has probability density function f given by
$$f ( x ) = \begin{cases} \frac { 3 } { 4 x ^ { 2 } } + \frac { 1 } { 4 } & 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
(i) Find the distribution function of $X$.\\
(ii) Find the exact value of the interquartile range of $X$.\\
\hfill \mbox{\textit{CAIE FP2 2019 Q7 [8]}}