| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2016 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Sketch or interpret PDF graph |
| Difficulty | Standard +0.3 This is a straightforward S2 question requiring differentiation of a piecewise CDF to obtain the PDF, sketching it, then computing E(X) by integration. While it involves multiple steps and careful handling of piecewise functions, these are standard techniques for this module with no novel problem-solving required. Slightly easier than average due to the mechanical nature of the calculus involved. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03e Find cdf: by integration5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| Part | Answer/Working | Marks |
| (a) | Candidate's own grid used is acceptable. | |
| Nothing drawn beyond 1 and 6 (or line at 0) | B1 | Must be at least one line drawn somewhere on the diagram |
| Differentiate to give \(f(x) = \frac{1}{4}\) for \(1 \leq x \leq 4\) | M1 | Allow this M1 if seen in (b); PI by correct line |
| Straight line joining (1, ¼) to (4, ¼) | A1 | Candidate's choice of scale. BOD this horizontal line. |
| \(f(x) = \frac{1}{8}(6 - x)\) for \(4 \leq x \leq 6\) | M1 | OE Allow this M1 if seen in (b); PI by correct line |
| Straight line joining (4, ¼) to (6, 0) | A1 | |
| Total: 5 | ||
| (b) | \(\int_1^4 \frac{1}{4} x \, dx\) and \(\int_4^6 \frac{1}{8} x(6-x) dx\) | M1 A1 |
| \(E(X) = \left[\frac{x^2}{8}\right]_1^4 + \left[\frac{3x^2}{8} - \frac{x^3}{24}\right]_4^6\) | A1 | Integrations done correctly and added |
| \(= 3\frac{1}{24}\) or \(\frac{73}{24}\) | A1 | Or AWRT 3.04 |
| Total: 4 | ||
| TOTAL FOR Q7: 9 |
| Part | Answer/Working | Marks | Guidance |
|------|---|---|---|
| (a) | Candidate's own grid used is acceptable. | | Accept wobbly lines as straight if it seems candidate has no ruler |
| | Nothing drawn beyond 1 and 6 (or line at 0) | B1 | Must be at least one line drawn somewhere on the diagram |
| | Differentiate to give $f(x) = \frac{1}{4}$ for $1 \leq x \leq 4$ | M1 | Allow this M1 if seen in (b); PI by correct line |
| | Straight line joining (1, ¼) to (4, ¼) | A1 | Candidate's choice of scale. BOD this horizontal line. |
| | $f(x) = \frac{1}{8}(6 - x)$ for $4 \leq x \leq 6$ | M1 | OE Allow this M1 if seen in (b); PI by correct line |
| | Straight line joining (4, ¼) to (6, 0) | A1 | |
| | **Total: 5** | | |
| (b) | $\int_1^4 \frac{1}{4} x \, dx$ and $\int_4^6 \frac{1}{8} x(6-x) dx$ | M1 A1 | For **both** integrands correct; Including correct limits for both |
| | $E(X) = \left[\frac{x^2}{8}\right]_1^4 + \left[\frac{3x^2}{8} - \frac{x^3}{24}\right]_4^6$ | A1 | Integrations done correctly and added |
| | $= 3\frac{1}{24}$ or $\frac{73}{24}$ | A1 | Or AWRT 3.04 |
| | **Total: 4** | | |
| | **TOTAL FOR Q7: 9** | | |The continuous random variable $X$ has a cumulative distribution function F($x$), where
$$\text{F}(x) = \begin{cases}
0 & x < 1 \\
\frac{1}{4}(x - 1) & 1 \leqslant x < 4 \\
\frac{1}{16}(12x - x^2 - 20) & 4 \leqslant x \leqslant 6 \\
1 & x > 6
\end{cases}$$
\begin{enumerate}[label=(\alph*)]
\item Sketch the probability density function, f($x$), on the grid below. [5 marks]
\item Find the mean value of $X$. [4 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA S2 2016 Q7 [9]}}