| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2010 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Piecewise PDF with multiple regions |
| Difficulty | Standard +0.3 This is a straightforward piecewise PDF question requiring standard integration techniques. Students must identify the linear functions from the graph, integrate to find probabilities and expectation, and solve for the median. While it involves multiple parts and careful setup of the piecewise function, these are routine S2 skills with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(0.5(0.5 + 0.75) \times 0.5\) or \(\int_1^{1.5} \frac{x}{2}\,dx\) | M1 | Attempt find correct area e.g. 1 squ + \(\frac{1}{4}\) squ or integral with correct limits any \(f(x)\) |
| \(= \frac{5}{16}\) or \(0.3125\) or \(0.313\) | A1 [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{1}{2}m \times \frac{m}{2}\) or \(\int_0^m \frac{x}{2}\,dx\) | M1 | Attempt area from 0 to \(m\) (or \(m\) to 2) their \(f(x)\) |
| \(= \frac{1}{2}\) | M1 | Expression for area \(= \frac{1}{2}\). Ignore limits |
| \(m = \sqrt{2}\) or \(1.41\) | A1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\int_0^2 \frac{x^2}{2}\,dx\) | M1 | Attempt \(\int x f(x)\,dx\). Ignore limits |
| \(= \frac{4}{3}\) oe | A1 [2] |
## Question 4:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $0.5(0.5 + 0.75) \times 0.5$ or $\int_1^{1.5} \frac{x}{2}\,dx$ | M1 | Attempt find correct area e.g. 1 squ + $\frac{1}{4}$ squ or integral with correct limits any $f(x)$ |
| $= \frac{5}{16}$ or $0.3125$ or $0.313$ | A1 [2] | |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{1}{2}m \times \frac{m}{2}$ or $\int_0^m \frac{x}{2}\,dx$ | M1 | Attempt area from 0 to $m$ (or $m$ to 2) their $f(x)$ |
| $= \frac{1}{2}$ | M1 | Expression for area $= \frac{1}{2}$. Ignore limits |
| $m = \sqrt{2}$ or $1.41$ | A1 [3] | |
### Part (iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int_0^2 \frac{x^2}{2}\,dx$ | M1 | Attempt $\int x f(x)\,dx$. Ignore limits |
| $= \frac{4}{3}$ oe | A1 [2] | |
---4\\
\includegraphics[max width=\textwidth, alt={}, center]{c7cbd61b-9a62-494a-b595-f624ec5c0bea-2_351_561_1562_794}
The diagram shows the graph of the probability density function, f , of a random variable $X$ which takes values between 0 and 2 only.\\
(i) Find $\mathrm { P } ( 1 < X < 1.5 )$.\\
(ii) Find the median of $X$.\\
(iii) Find $\mathrm { E } ( X )$.
\hfill \mbox{\textit{CAIE S2 2010 Q4 [7]}}