| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2013 |
| Session | January |
| Marks | 11 |
| 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 S2 question requiring standard techniques: sketching a piecewise PDF, finding k by integration (∫f=1), and calculating quartiles. The median is given by symmetry observation, and the lower quartile requires solving a single integral equation. All steps are routine applications of core PDF properties with no novel problem-solving required. |
| 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 |
4 A continuous random variable $X$ has probability density function defined by
$$f ( x ) = \begin{cases} k x ^ { 2 } & 0 \leqslant x \leqslant 3 \\ 9 k & 3 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$
\begin{enumerate}[label=(\alph*)]
\item Sketch the graph of f.
\item Show that the value of $k$ is $\frac { 1 } { 18 }$.
\item \begin{enumerate}[label=(\roman*)]
\item Write down the median value of $X$.
\item Calculate the value of the lower quartile of $X$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA S2 2013 Q4 [11]}}