| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| 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 S2 probability density function question requiring routine integration techniques. Parts (i)-(iii) involve direct application of standard formulas (normalisation, probability calculation, expectation), while part (iv) requires solving a cubic equation from the cumulative distribution function. All steps 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 pdf5.03c Calculate mean/variance: by integration5.03f Relate pdf-cdf: medians and percentiles |
A random variable $X$ has probability density function given by
$$f(x) = \begin{cases} k(1-x) & -1 \leq x \leq 1, \\ 0 & \text{otherwise}, \end{cases}$$
where $k$ is a constant.
\begin{enumerate}[label=(\roman*)]
\item Show that $k = \frac{1}{2}$. [2]
\item Find $\text{P}(X > \frac{1}{2})$. [1]
\item Find the mean of $X$. [3]
\item Find $a$ such that $\text{P}(X < a) = \frac{1}{3}$. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2011 Q7 [9]}}