| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2013 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Topic | Cumulative distribution functions |
| Type | PDF to CDF derivation |
| Difficulty | Standard +0.3 This is a standard S2 question requiring routine integration of simple polynomial functions to find CDFs and quartiles. The piecewise structure adds minimal complexity, and part (c)(i) is a 'show that' which guides students through the harder interval. All techniques are straightforward applications of core syllabus material 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 integration5.03e Find cdf: by integration5.03f Relate pdf-cdf: medians and percentiles |
7 A continuous random variable $X$ has the probability density function defined by
$$\mathrm { f } ( x ) = \left\{ \begin{array} { c c }
x ^ { 2 } & 0 \leqslant x \leqslant 1 \\
\frac { 1 } { 3 } ( 5 - 2 x ) & 1 \leqslant x \leqslant 2 \\
0 & \text { otherwise }
\end{array} \right.$$
\begin{enumerate}[label=(\alph*)]
\item Sketch the graph of f on the axes below.
\item \begin{enumerate}[label=(\roman*)]
\item Find the cumulative distribution function, F , for $0 \leqslant x \leqslant 1$.
\item Hence, or otherwise, find the value of the lower quartile of $X$.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Show that the cumulative distribution function for $1 \leqslant x \leqslant 2$ is defined by
$$\mathrm { F } ( x ) = \frac { 1 } { 3 } \left( 5 x - x ^ { 2 } - 3 \right)$$
\item Hence, or otherwise, find the value of the upper quartile of $X$.\\
\includegraphics[max width=\textwidth, alt={}, center]{03c1e107-3377-4b0d-9daf-7f70233c18b5-5_554_1050_1217_424}
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA S2 2013 Q7 [15]}}