| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2013 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Calculate and compare mean, median, mode |
| Difficulty | Standard +0.3 This is a standard S2 continuous probability distribution question requiring routine techniques: integrating to find k, differentiating to find the mode, and computing E(X) by integration. The algebra is straightforward (quadratic functions), and comparing mode vs mean for skewness is a textbook application. Slightly easier than average due to the simple polynomial form and standard multi-part structure. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\int f(x)\,dx = k\left[3x + x^2 - \frac{x^3}{3}\right]\) | M1 | For some correct integration \(x^n \to x^{n+1}\) for at least one term |
| \(\int_0^3 f(x)\,dx = 1\) gives \(k\left[\left(9+9-\frac{27}{3}\right)-(0)\right]=1\) | M1 | For correct use of limit 3 and at least implied use of limit 0, put \(=1\) |
| So \(k = \frac{1}{9}\) | A1cso | For correct solution with no incorrect working seen |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(f'(x) = k(2-2x)\) | M1 | For attempt to differentiate and put \(=0\); at least one correctly differentiated \(x\) term. Or completing the square / sketch and symmetry |
| \(f'(x)=0\) implies \(x=1\) so \(\text{mode} = 1\) | A1 | For mode \(= 1\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(E(X) = \int_0^3 \frac{1}{9}\left(3x + 2x^2 - x^3\right)dx\) | M1 | For clear attempt to use \(xf(x)\) with intention of integrating |
| \(= \frac{1}{9}\left[\frac{3x^2}{2}+\frac{2x^3}{3}-\frac{x^4}{4}\right]_0^3\) | M1dA1 | Dependent on 1st M; at least one correct term with correct coefficient |
| \(= \left\{\frac{1}{9}\left[\left(\frac{3}{2}\times 9 + \frac{2}{3}\times 27 - \frac{81}{4}\right)-0\right]\right\} = \frac{5}{4}\) | A1 | For answer \(\frac{5}{4}\) or 1.25 or exact equivalent |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Mean \(>\) mode | M1 | For a comparison of mean and mode (ft their values). Do not allow median |
| So positive skew | A1 | For positive skew only (provided compatible with their values and comparison) |
## Question 4:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int f(x)\,dx = k\left[3x + x^2 - \frac{x^3}{3}\right]$ | M1 | For some correct integration $x^n \to x^{n+1}$ for at least one term |
| $\int_0^3 f(x)\,dx = 1$ gives $k\left[\left(9+9-\frac{27}{3}\right)-(0)\right]=1$ | M1 | For correct use of limit 3 and at least implied use of limit 0, put $=1$ |
| So $k = \frac{1}{9}$ | A1cso | For correct solution with no incorrect working seen |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f'(x) = k(2-2x)$ | M1 | For attempt to differentiate and put $=0$; at least one correctly differentiated $x$ term. Or completing the square / sketch and symmetry |
| $f'(x)=0$ implies $x=1$ so $\text{mode} = 1$ | A1 | For mode $= 1$ |
### Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $E(X) = \int_0^3 \frac{1}{9}\left(3x + 2x^2 - x^3\right)dx$ | M1 | For clear attempt to use $xf(x)$ with intention of integrating |
| $= \frac{1}{9}\left[\frac{3x^2}{2}+\frac{2x^3}{3}-\frac{x^4}{4}\right]_0^3$ | M1dA1 | Dependent on 1st M; at least one correct term with correct coefficient |
| $= \left\{\frac{1}{9}\left[\left(\frac{3}{2}\times 9 + \frac{2}{3}\times 27 - \frac{81}{4}\right)-0\right]\right\} = \frac{5}{4}$ | A1 | For answer $\frac{5}{4}$ or 1.25 or exact equivalent |
### Part (d):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Mean $>$ mode | M1 | For a comparison of mean and mode (ft their values). Do **not** allow median |
| So **positive skew** | A1 | For positive skew only (provided compatible with their values and comparison) |
---4. The random variable $X$ has probability density function $\mathrm { f } ( x )$ given by
$$f ( x ) = \left\{ \begin{array} { c c }
k \left( 3 + 2 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 3 \\
0 & \text { otherwise }
\end{array} \right.$$
where $k$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Show that $k = \frac { 1 } { 9 }$
\item Find the mode of $X$.
\item Use algebraic integration to find $\mathrm { E } ( X )$.
By comparing your answers to parts (b) and (c),
\item describe the skewness of $X$, giving a reason for your answer.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2013 Q4 [11]}}