| Exam Board | WJEC |
|---|---|
| Module | Further Unit 2 (Further Unit 2) |
| Year | 2024 |
| Session | June |
| Marks | 13 |
| 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 piecewise CDF question requiring standard techniques: finding percentiles by solving F(x) = 0.75, differentiating to get the pdf, and computing E(X) by integration. All steps are routine for Further Maths students with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration5.03e Find cdf: by integration5.03f Relate pdf-cdf: medians and percentiles |
2. Emlyn aims to produce podcast episodes that are a standard length of time, which he calls the 'target time'. The time, $X$ minutes, above or below the target time, which he calls the 'allowed time', can be modelled by the following cumulative distribution function.
$$F ( x ) = \begin{cases} 0 & x < - 2 \\ \frac { x + 2 } { 5 } & - 2 \leqslant x < 1 \\ \frac { x ^ { 2 } - x + 3 } { 5 } & 1 \leqslant x \leqslant 2 \\ 1 & x > 2 \end{cases}$$
\begin{enumerate}[label=(\alph*)]
\item Calculate the upper quartile for the 'allowed time'.
\item Find $f ( x )$, the probability density function, for all values of $x$.
\item \begin{enumerate}[label=(\roman*)]
\item Calculate the mean 'allowed time'.
\item Interpret your answer in context.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 2 2024 Q2 [13]}}