| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2011 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Find median or percentiles |
| Difficulty | Standard +0.3 This is a straightforward S2 question requiring standard techniques: verifying probabilities by integration, identifying quartiles from given probabilities, and applying conditional probability. The piecewise pdf is simple, calculations are routine, and part (c) requires only recognition that the given probabilities correspond to quartiles. Slightly easier than average due to the 'show that' structure providing answers to verify. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
6 The continuous random variable $X$ has probability density function defined by
$$\mathrm { f } ( x ) = \begin{cases} \frac { 3 } { 8 } x ^ { 2 } & 0 \leqslant x \leqslant \frac { 1 } { 2 } \\ \frac { 3 } { 32 } & \frac { 1 } { 2 } \leqslant x \leqslant 11 \\ 0 & \text { otherwise } \end{cases}$$
\begin{enumerate}[label=(\alph*)]
\item Sketch the graph of f.
\item Show that:
\begin{enumerate}[label=(\roman*)]
\item $\quad \mathrm { P } \left( X \geqslant 8 \frac { 1 } { 3 } \right) = \frac { 1 } { 4 }$;
\item $\quad \mathrm { P } ( X \geqslant 3 ) = \frac { 3 } { 4 }$.
\end{enumerate}\item Hence write down the exact value of:
\begin{enumerate}[label=(\roman*)]
\item the interquartile range of $X$;
\item the median, $m$, of $X$.
\end{enumerate}\item Find the exact value of $\mathrm { P } ( X < m \mid X \geqslant 3 )$.
\end{enumerate}
\hfill \mbox{\textit{AQA S2 2011 Q6 [12]}}