| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2013 |
| Session | January |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Find multiple parameters from system |
| Difficulty | Standard +0.3 This is a standard S2 probability density function question requiring systematic application of well-known properties: integrating to 1 for part (a), using the expectation formula for part (b), solving simultaneous equations for part (c), integrating to find the median for part (d), and comparing mean/median for skewness in part (e). All techniques are routine for S2 students with no novel insight required, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\int_0^5 a+bx\, dx = 1\) | M1 | Attempting to integrate with correct limits, or attempt to find area \(0.5(a+b)h\) |
| \(\left[ax + \frac{bx^2}{2}\right]_0^5 = 1\) | A1 | Correct integration or correct area |
| \(5a + \frac{25b}{2} = 1\), giving \(10a + 25b = 2\) | M1dep, A1cso | 2nd M1 for using \(=1\), dependent on first M1. Condone missing \(dx\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\int_0^5 ax+bx^2\, dx = \frac{35}{12}\) | M1 | Using/writing this expression (limits not needed) |
| \(\left[\frac{ax^2}{2} + \frac{bx^3}{3}\right]_0^5 = \frac{35}{12}\) | A1 | Correct integration |
| \(\frac{25a}{2} + \frac{125b}{3} = \frac{35}{12}\), giving \(30a + 100b = 7\) | A1 | May be awarded for unsimplified version |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Solve \(30a+100b=7\) and \(10a+25b=2\) simultaneously | M1 | Either rearranging and substitution or making coefficients same |
| \(a=0.1\), \(b=0.04\) | A1, A1 | 1st A1 for 0.1, 2nd A1 for 0.04 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\int_0^m 0.1 + 0.04x\, dx = 0.5\) | M1 | Writing/using with their \(a\) and \(b\), limits not needed |
| \(\left[0.1x + \frac{0.04x^2}{2}\right]_0^m = 0.5\) | A1ft | Correct integration for their \(a\) and \(b\) |
| \(0.1m + 0.02m^2 - 0.5 = 0\) (simplifies to \(m^2+m-25=0\)) | NB correct equation simplifies to \(m^2+m-25=0\) | |
| \(m = \frac{-0.1 \pm \sqrt{0.1^2 + 4\times0.02\times0.5}}{2\times0.02}\) | ||
| \(m = 3.09\), \(-8.09\), therefore \(m=3.09\) | A1 | 3.09 only. If both roots given must select 3.09 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| mean \(<\) median (\(<\) mode) | B1ft | Must compare their values for mean and median correctly |
| negatively skewed | B1 dep ft | Dependent on previous B. Only need to compare 2 of mean, median, mode |
## Question 7:
**(a)**
| Answer | Mark | Guidance |
|--------|------|----------|
| $\int_0^5 a+bx\, dx = 1$ | M1 | Attempting to integrate with correct limits, or attempt to find area $0.5(a+b)h$ |
| $\left[ax + \frac{bx^2}{2}\right]_0^5 = 1$ | A1 | Correct integration or correct area |
| $5a + \frac{25b}{2} = 1$, giving $10a + 25b = 2$ | M1dep, A1cso | 2nd M1 for using $=1$, dependent on first M1. Condone missing $dx$ |
**(b)**
| Answer | Mark | Guidance |
|--------|------|----------|
| $\int_0^5 ax+bx^2\, dx = \frac{35}{12}$ | M1 | Using/writing this expression (limits not needed) |
| $\left[\frac{ax^2}{2} + \frac{bx^3}{3}\right]_0^5 = \frac{35}{12}$ | A1 | Correct integration |
| $\frac{25a}{2} + \frac{125b}{3} = \frac{35}{12}$, giving $30a + 100b = 7$ | A1 | May be awarded for unsimplified version |
**(c)**
| Answer | Mark | Guidance |
|--------|------|----------|
| Solve $30a+100b=7$ and $10a+25b=2$ simultaneously | M1 | Either rearranging and substitution or making coefficients same |
| $a=0.1$, $b=0.04$ | A1, A1 | 1st A1 for 0.1, 2nd A1 for 0.04 |
**(d)**
| Answer | Mark | Guidance |
|--------|------|----------|
| $\int_0^m 0.1 + 0.04x\, dx = 0.5$ | M1 | Writing/using with their $a$ and $b$, limits not needed |
| $\left[0.1x + \frac{0.04x^2}{2}\right]_0^m = 0.5$ | A1ft | Correct integration for their $a$ and $b$ |
| $0.1m + 0.02m^2 - 0.5 = 0$ (simplifies to $m^2+m-25=0$) | | NB correct equation simplifies to $m^2+m-25=0$ |
| $m = \frac{-0.1 \pm \sqrt{0.1^2 + 4\times0.02\times0.5}}{2\times0.02}$ | | |
| $m = 3.09$, $-8.09$, therefore $m=3.09$ | A1 | 3.09 only. If both roots given must select 3.09 |
**(e)**
| Answer | Mark | Guidance |
|--------|------|----------|
| mean $<$ median ($<$ mode) | B1ft | Must compare their values for mean and median correctly |
| negatively skewed | B1 dep ft | Dependent on previous B. Only need to compare 2 of mean, median, mode |7. The continuous random variable $X$ has the following probability density function
$$f ( x ) = \begin{cases} a + b x & 0 \leqslant x \leqslant 5 \\ 0 & \text { otherwise } \end{cases}$$
where $a$ and $b$ are constants.
\begin{enumerate}[label=(\alph*)]
\item Show that $10 a + 25 b = 2$
Given that $\mathrm { E } ( X ) = \frac { 35 } { 12 }$
\item find a second equation in $a$ and $b$,
\item hence find the value of $a$ and the value of $b$.
\item Find, to 3 significant figures, the median of $X$.
\item Comment on the skewness. Give a reason for your answer.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2013 Q7 [15]}}