| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2012 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Single-piece PDF with k |
| Difficulty | Moderate -0.3 This is a standard S2 probability density function question requiring routine integration to find k, solving an equation for a percentile, and interpreting a graph for the median. Part (i) is straightforward integration with the total probability condition, part (ii) involves setting up and solving ∫f(x)dx = 1/5, and part (iii) requires only visual interpretation of symmetry/skewness. All techniques are textbook exercises 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 pdf5.03e Find cdf: by integration5.03f Relate pdf-cdf: medians and percentiles |
**(i)**
**M1** Any attempt integ $f(x)$ & $= 1$. Ignore limits
**A1** $\int_0^1 \frac{k}{(x+1)^2} dx = 1$; $\left[-\frac{k}{x+1}\right]_0^1 = 1$; $-k\left(\frac{1}{2}\right) - (-1) = \frac{1}{2}$
**[2]** oe, with limits inserted correctly
$(k = 2$ AG$)$
**(ii)**
**M1** Attempt integ $f(x)$ & $= 1$ (oe), ignore limits
**A1** $\int_0^a \frac{2}{(x+1)^2} dx = \frac{1}{5}$
**A1** $\left[-\frac{2}{x+1}\right]_0^a = \frac{1}{5}$; $-\frac{2}{a+1} - (-2) = \frac{1}{5}$
**[3]** oe, with correct limits inserted correctly
$a = 9$
**(iii)**
**B1** Area below $x = 0.5$ is greater than $0.5$
**B1dep** $m < 0.5$
**[2]** oe, eg More area at left hand end
---4 The random variable $X$ has probability density function given by
$$f ( x ) = \begin{cases} \frac { k } { ( x + 1 ) ^ { 2 } } & 0 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$
where $k$ is a constant.\\
(i) Show that $k = 2$.\\
(ii) Find $a$ such that $\mathrm { P } ( X < a ) = \frac { 1 } { 5 }$.\\
(iii)\\
\includegraphics[max width=\textwidth, alt={}, center]{18cef198-5ca2-4700-88e9-1a2bd55f841e-2_367_524_1548_849}
The diagram shows the graph of $y = \mathrm { f } ( x )$. The median of $X$ is denoted by $m$. Use the diagram to explain whether $m < 0.5$, $m = 0.5$ or $m > 0.5$.
\hfill \mbox{\textit{CAIE S2 2012 Q4 [7]}}