| Exam Board | OCR |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2011 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Piecewise PDF with k |
| Difficulty | Moderate -0.3 This is a standard S3 piecewise PDF question requiring routine integration to find the constant, probability calculation, and expectation. While it involves multiple parts and careful handling of the piecewise structure, the techniques are straightforward applications of core probability theory with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int_0^1 audx + \int_1^2 a(x-2)^2dx = 1\) | M1 | |
| \(\left[\frac{ax^2}{2}\right]_0^1 + \left[\frac{a(x-2)^3}{3}\right]_1^2\) | B1 | With or without limits |
| \(\frac{1}{2}a + \frac{1}{3}a = 1\) | M1 | |
| \(a = \frac{6}{5}\) | A1 | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| EITHER: \(\int_0^a audx + \int_1^{1.5} a(x-2)^2dx\) | M1 | Any \(a\) |
| OR \(1 - \int_{-2}^{1.5} a(x-2)^2dx\) | ||
| \(= \frac{19}{20}\) | A1 | 2 AEF |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int_1^2 u^2dx + \int_2^* a(x(x-2)^2dx\) | M1 | |
| \(\left[\frac{ax^3}{3}\right]_0^1 + \left[a\left(\frac{x^4}{4} - \frac{4x^3}{3} + 2x^2\right)\right]_1^2\) | B1 | AEF With or without limits |
| \(=\frac{9}{10}\) | A1 | 3(9) AEF |
| (Expected monthly demand = 900) |
**3(i)**
$\int_0^1 audx + \int_1^2 a(x-2)^2dx = 1$ | M1 |
$\left[\frac{ax^2}{2}\right]_0^1 + \left[\frac{a(x-2)^3}{3}\right]_1^2$ | B1 | With or without limits
$\frac{1}{2}a + \frac{1}{3}a = 1$ | M1 |
$a = \frac{6}{5}$ | A1 | 4
**3(ii)**
EITHER: $\int_0^a audx + \int_1^{1.5} a(x-2)^2dx$ | M1 | Any $a$
OR $1 - \int_{-2}^{1.5} a(x-2)^2dx$ | |
$= \frac{19}{20}$ | A1 | 2 AEF
**3(iii)**
$\int_1^2 u^2dx + \int_2^* a(x(x-2)^2dx$ | M1 |
$\left[\frac{ax^3}{3}\right]_0^1 + \left[a\left(\frac{x^4}{4} - \frac{4x^3}{3} + 2x^2\right)\right]_1^2$ | B1 | AEF With or without limits
$=\frac{9}{10}$ | A1 | 3(9) AEF
(Expected monthly demand = 900) | |
---3 The monthly demand for a product, $X$ thousand units, is modelled by the random variable $X$ with probability density function given by
$$f ( x ) = \begin{cases} a x & 0 \leqslant x \leqslant 1 \\ a ( x - 2 ) ^ { 2 } & 1 < x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$
where $a$ is a positive constant. Find\\
(i) the value of $a$,\\
(ii) the probability that the monthly demand is at most 1500 units,\\
(iii) the expected monthly demand.
\hfill \mbox{\textit{OCR S3 2011 Q3 [9]}}