| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2014 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Derive or verify variance formula |
| Difficulty | Standard +0.3 This is a structured, multi-part question on continuous uniform distributions that guides students through standard derivations. Parts (a)(i)-(iii) involve routine integration and applying definitions (normalizing pdf, finding E(X²)), while part (a)(iv) uses Var(X)=E(X²)-[E(X)]² with given expressions. Part (b) is straightforward algebra with specific values. The question requires careful algebraic manipulation but no novel insight—it's slightly easier than average because of its highly scaffolded structure and standard S2 content. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
I'd be happy to help clean up the mark scheme content, but I don't see any actual mark scheme text in your message. You've provided "Question 4: 4" but there's no detailed marking content with annotations like M1, A1, B1, etc. to convert.
Please provide the full extracted mark scheme content that needs cleaning up, and I'll convert the unicode symbols to LaTeX notation and format it clearly.4 A continuous random variable $X$ has a probability density function defined by
$$f ( x ) = \begin{cases} \frac { 1 } { k } & a \leqslant x \leqslant b \\ 0 & \text { otherwise } \end{cases}$$
where $b > a > 0$.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Prove that $k = b - a$.
\item Write down the value of $\mathrm { E } ( X )$.
\item Show, by integration, that $\mathrm { E } \left( X ^ { 2 } \right) = \frac { 1 } { 3 } \left( b ^ { 2 } + a b + a ^ { 2 } \right)$.
\item Hence derive a simplified formula for $\operatorname { Var } ( X )$.
\end{enumerate}\item Given that $a = 4$ and $\operatorname { Var } ( X ) = 3$, find the numerical value of $\mathrm { E } ( X )$.
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{34517557-011e-4956-be7d-b26fe5e64d0a-08_1347_1707_1356_153}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{AQA S2 2014 Q4 [11]}}