| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2010 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Find constant k in PDF |
| Difficulty | Moderate -0.8 This is a straightforward rectangular/uniform distribution question requiring only standard techniques: sketching a constant pdf, using the integral-equals-one property to find k, applying standard formulas for mean and variance of uniform distributions, and basic probability calculations. All steps are routine S2 content with no problem-solving insight required, making it easier than average but not trivial due to the algebraic manipulation with parameter k. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
The continuous random variable $X$ has a rectangular distribution defined by
$$f(x) = \begin{cases} k & -3k \leqslant x \leqslant k \\ 0 & \text{otherwise} \end{cases}$$
\begin{enumerate}[label=(\alph*)]
\item
\begin{enumerate}[label=(\roman*)]
\item Sketch the graph of f. [2 marks]
\item Hence show that $k = \frac{1}{2}$. [2 marks]
\end{enumerate}
\item Find the exact numerical values for the mean and the standard deviation of $X$. [3 marks]
\item
\begin{enumerate}[label=(\roman*)]
\item Find $\mathrm{P}\left(X \geqslant -\frac{1}{4}\right)$. [2 marks]
\item Write down the value of $\mathrm{P}\left(X \neq -\frac{1}{4}\right)$. [1 mark]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{AQA S2 2010 Q3 [10]}}