| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2011 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Find constant k in PDF |
| Difficulty | Moderate -0.8 This is a straightforward S2 question testing basic PDF properties. Parts (ii) and (iii) involve routine integration (∫a/x² dx and finding E(X)), part (i) is a simple sketch, and part (iv) tests understanding of uniform distributions. All techniques are standard with no problem-solving insight required, making it easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| (i) | B1 | Horizontal line above axis |
| B1 | Concave decreasing curve above axis | |
| B1 | Both correct including approx relationship, not extending beyond [1, 3], verticals and scale not needed | |
| 3 | ||
| (ii) \(\int_a^1 f_2(x) dx = 1\), \(\left[-\frac{a}{x}\right]_a^1 = 1; a = \frac{3}{2}\) | M1 | Attempt \(\int f_2(x)dx\), limits [1, 3] at some stage, equate to 1 |
| B1 | Correct indefinite integral | |
| A1 | Correctly obtain 3/2 or 1.5 or exact equivalent | |
| 3 | ||
| (iii) \(\int_1^3 x f_3(x) dx = \left[a \ln x\right]\) | M1 | Attempt \(\int xf_3(x)dx\), limits [1, 3] at some stage |
| B1 FT | Correct indefinite integral, FT on a | |
| A1 FT 3 | Answer, exactly equivalent or a.r.t. 1.65, FT on a or \(\ln 3\) | |
| 3 |
**(i)** | B1 | Horizontal line above axis
| B1 | Concave decreasing curve above axis
| B1 | Both correct including approx relationship, not extending beyond [1, 3], verticals and scale not needed
| | 3
**(ii)** $\int_a^1 f_2(x) dx = 1$, $\left[-\frac{a}{x}\right]_a^1 = 1; a = \frac{3}{2}$ | M1 | Attempt $\int f_2(x)dx$, limits [1, 3] at some stage, equate to 1
| B1 | Correct indefinite integral
| A1 | Correctly obtain 3/2 or 1.5 or exact equivalent
| | 3
**(iii)** $\int_1^3 x f_3(x) dx = \left[a \ln x\right]$ | M1 | Attempt $\int xf_3(x)dx$, limits [1, 3] at some stage
| B1 FT | Correct indefinite integral, FT on a
| A1 FT 3 | Answer, exactly equivalent or a.r.t. 1.65, FT on a or $\ln 3$
| | 37 Two continuous random variables $S$ and $T$ have probability density functions $\mathrm { f } _ { S }$ and $\mathrm { f } _ { T }$ given respectively by
$$\begin{aligned}
& f _ { S } ( x ) = \begin{cases} \frac { a } { x ^ { 2 } } & 1 \leqslant x \leqslant 3 \\
0 & \text { otherwise } \end{cases} \\
& f _ { T } ( x ) = \begin{cases} b & 1 \leqslant x \leqslant 3 \\
0 & \text { otherwise } \end{cases}
\end{aligned}$$
where $a$ and $b$ are constants.\\
(i) Sketch on the same axes the graphs of $y = \mathrm { f } _ { S } ( x )$ and $y = \mathrm { f } _ { T } ( x )$.\\
(ii) Find the value of $a$.\\
(iii) Find $\mathrm { E } ( S )$.\\
(iv) A student gave the following description of the distribution of $T$ : "The probability that $T$ occurs is constant". Give an improved description, in everyday terms.
\hfill \mbox{\textit{OCR S2 2011 Q7 [10]}}