| Exam Board | OCR |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2007 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Geometric/graphical PDF with k |
| Difficulty | Moderate -0.3 This is a straightforward S3 question requiring standard techniques: using the total probability property (area = 1) to find k, determining line equations from a graph, writing the piecewise PDF, and calculating E(X) by integration. All steps are routine applications of core probability concepts with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use \(3a/2 = 1\) | B1 | 1 Or similar |
| (ii) \(y = \frac{2}{3}x\); \(y = 1 - \frac{1}{3}x\) | B1, M1A1 | 3 M1 for correct gradient; B1M1A0 if not \(y = ...\) |
| Answer | Marks | Guidance |
|---|---|---|
| B1√ | 1 ft (ii) | |
| (iv) \(\int_0^1 \frac{2}{3}x^2 dx + \int_1^3 (x - \frac{1}{3}x^2)dx\) | M1 | One correct, with limits |
| \[\left[\frac{2}{9}x^3\right]_0^1 + \left[\frac{1}{2}x^2 - \frac{1}{9}x^3\right]_1^3 = \frac{4}{3}\] | A1√A1√, A1 | ft from similar \(f\); 4 aef |
**(i)** Use $3a/2 = 1$ | B1 | 1 Or similar
**(ii)** $y = \frac{2}{3}x$; $y = 1 - \frac{1}{3}x$ | B1, M1A1 | 3 M1 for correct gradient; B1M1A0 if not $y = ...$
**(iii)**
$$f(x) = \begin{cases}
\frac{2}{3}x & 0 \le x \le 1 \\
1 - \frac{1}{3}x & 1 < x \le 3
\end{cases}$$
| B1√ | 1 ft (ii)
**(iv)** $\int_0^1 \frac{2}{3}x^2 dx + \int_1^3 (x - \frac{1}{3}x^2)dx$ | M1 | One correct, with limits
$$\left[\frac{2}{9}x^3\right]_0^1 + \left[\frac{1}{2}x^2 - \frac{1}{9}x^3\right]_1^3 = \frac{4}{3}$$ | A1√A1√, A1 | ft from similar $f$; 4 aef2 The continuous random variable $X$ takes values in the interval $0 \leqslant x \leqslant 3$ only with probability density function f . The graph of $y = \mathrm { f } ( x )$ consists of the two line segments shown in the diagram.\\
\includegraphics[max width=\textwidth, alt={}, center]{4a6d94a2-66e1-449a-ac0e-1fbada74bb3b-2_524_1287_950_429}\\
(i) Show that $a = \frac { 2 } { 3 }$.\\
(ii) Find the equations of the two line segments.\\
(iii) Hence write down the probability density function of $X$.\\
(iv) Find $\mathrm { E } ( X )$.
\hfill \mbox{\textit{OCR S3 2007 Q2 [9]}}