| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2002 |
| Session | June |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Sketch or interpret PDF graph |
| Difficulty | Moderate -0.3 This is a standard S2 question on piecewise probability density functions requiring routine integration and CDF construction. While it has multiple parts and requires careful bookkeeping across intervals, all techniques are straightforward applications of definitions with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03c Calculate mean/variance: by integration5.03e Find cdf: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| - Horizontal base from \((10,0)\) extending right | B1 (labels); B1 (graph); B1 (axes) | Check: axes labelled; correct shape; correct scale markings |
| Answer | Marks |
|---|---|
| (i) \(F(x) = \int_0^x \frac{x}{15} dx = \frac{x^2}{30}\) for \(0 \leq x \leq 2\) | B1 |
| (ii) \(F(x) = \frac{12}{15} + \int_7^x \left(\frac{4}{9} - \frac{2x}{45}\right)dx = \frac{4x}{9} - \frac{x^2}{45} - \frac{11}{9}\) for \(7 \leq x \leq 10\) | B1 M1 A1 |
| (iii) \(F(x) = \frac{2}{15} + \int_2^x \frac{2}{15} dx = \frac{2x}{15} - \frac{2}{15}\) for \(2 \leq x \leq 7\) | B1 M1 A1 |
| (iv) \(F(x) = 0, x < 0\), \(F(x) = 1, x > 10\) | B1 (8 marks total) |
| Answer | Marks |
|---|---|
| \(P(X \leq 8.2) = F(8.2) = 0.928\) | M1 A1 (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(E(X) = \int_0^2 \frac{x^2}{15} dx + \int_2^7 \frac{2x}{15} dx + \int_7^{10} \left(\frac{4x}{9} - \frac{2x^2}{45}\right)dx\) | M1 A1 | |
| \(= \left[\frac{x^3}{45}\right]_0 + \left[\frac{x^2}{15}\right]_2 + \left[\frac{2x^2}{9} - \frac{2x^3}{125}\right]_7^{10} = 4.78\) | A1 A1 (4 marks) | Marks shown for: M1 (setting up integral), A1 (antiderivatives correct), A1 A1 (evaluation and final answer) |
## Part (a)
Graph shows a trapezoid/trapezium with:
- Horizontal base from $(0,0)$ to $(2,0)$
- Vertical line from $(2,0)$ to $(2, \frac{2}{15})$
- Horizontal top from $(2, \frac{2}{15})$ to $(7, \frac{2}{15})$
- Sloped line from $(7, \frac{2}{15})$ to $(10,0)$
- Horizontal base from $(10,0)$ extending right | B1 (labels); B1 (graph); B1 (axes) | Check: axes labelled; correct shape; correct scale markings
## Part (b)
**(i)** $F(x) = \int_0^x \frac{x}{15} dx = \frac{x^2}{30}$ for $0 \leq x \leq 2$ | B1
**(ii)** $F(x) = \frac{12}{15} + \int_7^x \left(\frac{4}{9} - \frac{2x}{45}\right)dx = \frac{4x}{9} - \frac{x^2}{45} - \frac{11}{9}$ for $7 \leq x \leq 10$ | B1 M1 A1
**(iii)** $F(x) = \frac{2}{15} + \int_2^x \frac{2}{15} dx = \frac{2x}{15} - \frac{2}{15}$ for $2 \leq x \leq 7$ | B1 M1 A1
**(iv)** $F(x) = 0, x < 0$, $F(x) = 1, x > 10$ | B1 (8 marks total)
## Part (c)
$P(X \leq 8.2) = F(8.2) = 0.928$ | M1 A1 (2 marks)
## Part (d)
$E(X) = \int_0^2 \frac{x^2}{15} dx + \int_2^7 \frac{2x}{15} dx + \int_7^{10} \left(\frac{4x}{9} - \frac{2x^2}{45}\right)dx$ | M1 A1
$= \left[\frac{x^3}{45}\right]_0 + \left[\frac{x^2}{15}\right]_2 + \left[\frac{2x^2}{9} - \frac{2x^3}{125}\right]_7^{10} = 4.78$ | A1 A1 (4 marks) | Marks shown for: M1 (setting up integral), A1 (antiderivatives correct), A1 A1 (evaluation and final answer)The continuous random variable $X$ has probability density function
$$f(x) = \begin{cases}
\frac{x}{15}, & 0 \leq x \leq 2, \\
\frac{2}{15}, & 2 < x < 7, \\
\frac{4}{9} - \frac{2x}{45}, & 7 \leq x \leq 10, \\
0, & \text{otherwise}.
\end{cases}$$
\begin{enumerate}[label=(\alph*)]
\item Sketch $f(x)$ for all values of $x$. [3]
\item \begin{enumerate}[label=(\roman*)]
\item Find expressions for the cumulative distribution function, $\mathrm{F}(x)$, for $0 \leq x \leq 2$ and for $7 \leq x \leq 10$.
\item Show that for $2 < x < 7$, $\mathrm{F}(x) = \frac{2x}{15} - \frac{2}{15}$.
\item Specify $\mathrm{F}(x)$ for $x < 0$ and for $x > 10$.
\end{enumerate} [8]
\item Find $\mathrm{P}(X \leq 8.2)$. [2]
\item Find, to 3 significant figures, $\mathrm{E}(X)$. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2002 Q7 [17]}}