| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2019 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Find or specify CDF |
| Difficulty | Standard +0.3 This is a straightforward application of standard techniques for continuous probability distributions. Part (i) requires integrating the pdf to find the cdf, which involves a simple polynomial and reciprocal integration. Part (ii) requires solving F(x) = 0.25 and F(x) = 0.75, leading to quadratic equations. While there are multiple steps and some algebraic manipulation, these are routine procedures covered extensively in Further Maths statistics with no novel problem-solving required. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03e Find cdf: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| 7(i) | F(x) = ∫ f(x) dx = – 3/(4x) + x/4 [+ c] | M1 |
| = – 3/(4x) + x/4 + ½ or ¼ (– 3/x + x + 2) | A1 | using F(1) = 0 or F(3) = 1 to find c if necessary |
| F(x) = 0 (x < or ≤1), F(x) = 1 (x > or≥3) | A1 | State F(x) for other values of x |
| Answer | Marks | Guidance |
|---|---|---|
| 7(ii) | ∫ Q f(x) dx = – 3/4Q + Q/4 + ½ = ¼ [or ¾] (AEF) | |
| 1 | M1 | Formulate equation for either quartile value Q |
| Q2 + [or – ] Q – 3 = 0 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| 1 3 | A1 A1 | Find lower quartile Q and upper quartile Q |
| Answer | Marks | Guidance |
|---|---|---|
| 3 1 | A1 | Find interquartile range |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 7:
--- 7(i) ---
7(i) | F(x) = ∫ f(x) dx = – 3/(4x) + x/4 [+ c] | M1 | Find or state distribution function F(x) for 1 ≤x≤3
= – 3/(4x) + x/4 + ½ or ¼ (– 3/x + x + 2) | A1 | using F(1) = 0 or F(3) = 1 to find c if necessary
F(x) = 0 (x < or ≤1), F(x) = 1 (x > or≥3) | A1 | State F(x) for other values of x
3
--- 7(ii) ---
7(ii) | ∫ Q f(x) dx = – 3/4Q + Q/4 + ½ = ¼ [or ¾] (AEF)
1 | M1 | Formulate equation for either quartile value Q
Q2 + [or – ] Q – 3 = 0 | A1
Q = ½ (– 1 + √13), Q = ½ (1 + √13)
1 3 | A1 A1 | Find lower quartile Q and upper quartile Q
1 3
Q – Q = 1
3 1 | A1 | Find interquartile range
5
Question | Answer | Marks | GuidanceThe continuous random variable $X$ has probability density function f given by
$$f(x) = \begin{cases}
\frac{3}{4x^2} + \frac{1}{4} & 1 \leqslant x \leqslant 3, \\
0 & \text{otherwise}.
\end{cases}$$
(i) Find the distribution function of $X$. [3]
(ii) Find the exact value of the interquartile range of $X$. [5]
\hfill \mbox{\textit{CAIE FP2 2019 Q7 [8]}}