| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2011 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Verify geometric PDF from graph |
| Difficulty | Moderate -0.3 This is a standard S2 question on continuous probability distributions requiring routine techniques: deriving pdf from a graph, integrating to find cdf, and finding median/mode. While multi-part with 13 marks total, each component follows textbook procedures with no novel problem-solving required. The triangular distribution is straightforward, making this slightly easier than average A-level maths questions. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03e Find cdf: by integration5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(m = \frac{4}{0.5} = -8\); \(f(x) = 4 - 8x\) (*) | M1, A1cso | |
| \[f(x) = \begin{cases} -8x + 4 & 0 \leq x \leq 0.5 \\ 0 & \text{otherwise} \end{cases}\] | B1, B1 | (4 marks) |
| (b) \(F(x) = \int_0^x (-8x + 4)dx\); \(F(x) = \left[-4x^2 + 4x\right]_0^x\) | M1, M1 | |
| \[F(x) = \begin{cases} 0 & x < 0 \\ -4x^2 + 4x & 0 \leq x \leq 0.5 \\ 1 & x > 0.5 \end{cases}\] | A1, B1 | (4 marks) |
| (c) \(-4x^2 + 4x = 0.5\); \(x = \frac{1}{4}(2 - \sqrt{2}) = 0.146\) | M1, M1A1 | (3 marks) |
| (d) \(x = 0\) | B1 | (1 mark) |
| (e) Positive Skew as mode<median | B1ft | (1 mark) |
**(a)** $m = \frac{4}{0.5} = -8$; $f(x) = 4 - 8x$ (*) | M1, A1cso |
$$f(x) = \begin{cases} -8x + 4 & 0 \leq x \leq 0.5 \\ 0 & \text{otherwise} \end{cases}$$ | B1, B1 | (4 marks)
**(b)** $F(x) = \int_0^x (-8x + 4)dx$; $F(x) = \left[-4x^2 + 4x\right]_0^x$ | M1, M1 |
$$F(x) = \begin{cases} 0 & x < 0 \\ -4x^2 + 4x & 0 \leq x \leq 0.5 \\ 1 & x > 0.5 \end{cases}$$ | A1, B1 | (4 marks)
**(c)** $-4x^2 + 4x = 0.5$; $x = \frac{1}{4}(2 - \sqrt{2}) = 0.146$ | M1, M1A1 | (3 marks)
**(d)** $x = 0$ | B1 | (1 mark)
**(e)** Positive Skew as mode<median | B1ft | (1 mark)
**Notes:**
- (a) M1 for $\pm\frac{4}{0.5}$ or attempt at gradient; A1 cso for proceeding to given expression with no incorrect working seen. B1 for top line. Must have $f(x)$ and $\{$ and more than one line. Condone use of <. B1 for 0 otherwise and no other parts.
- (b) M1 attempting to integrate (at least one $x^n \to x^{n+1}$) (ignore limits); M1 correct limits used or +C and either $F(0) = 0$ or $F(0.5) = 1$, may be implied by seeing $4x - 4x^2$; A1 middle line. May write $4x - 4x^2$. B1 top and bottom line.
- (c) M1 Their $F(x) = 0.5$; M1 attempting to solve – either correct use of quadratic formula or correct completion of the square; A1 awrt 0.146 or $\frac{2-\sqrt{2}}{4}$ o.e.
- (d) B1 for 0.
- (e) B1 ft their mode and median. Need direction and correct corresponding reason. OR B1 positive skew from tail on right hand side in diagram.
**[13 marks total]**
---A continuous random variable $X$ has the probability density function f($x$) shown in Figure 1.
\includegraphics{figure_1}
Figure 1
\begin{enumerate}[label=(\alph*)]
\item Show that f($x$) = $4 - 8x$ for $0 \leqslant x \leqslant 0.5$ and specify f($x$) for all real values of $x$.
[4]
\item Find the cumulative distribution function F($x$).
[4]
\item Find the median of $X$.
[3]
\item Write down the mode of $X$.
[1]
\item State, with a reason, the skewness of $X$.
[1]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2011 Q5 [13]}}