| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2002 |
| Session | January |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Cumulative distribution functions |
| Type | Continuous CDF with polynomial pieces |
| Difficulty | Standard +0.3 This is a standard S2 CDF question requiring routine techniques: finding k from F(2)=1, solving a quadratic for the median, differentiating for the pdf, and computing E(X) by integration. All steps are textbook procedures with no novel insight 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 integration5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| \(8k = 1,\; k = \frac{1}{8}\) | B1 | cso |
| Answer | Marks | Guidance |
|---|---|---|
| \(F(m) = 0.5\) | M1 | |
| \(x^2 + 2x - 4 = 0\) | A1 | |
| \(x = \sqrt{5} - 1 = 1.236\) | A1 | awrt 1.24 |
| Answer | Marks | Guidance |
|---|---|---|
| \(f(x) = \frac{1}{4}(x+1),\quad 0 \leq x \leq 2\) | M1A1 | Differentiation, all correct |
| \(f(x) = 0\), otherwise | A1 | 0 and ranges |
| Answer | Marks | Guidance |
|---|---|---|
| Graph of \(f(x)\): linear from \(\frac{1}{4}\) at \(x=0\) to \(\frac{3}{4}\) at \(x=2\), zero elsewhere | B1, B1, B1 | Values & labels; slope; \(f(x)=0\) |
| Answer | Marks |
|---|---|
| mode \(= 2\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(E(X) = \int_0^2 x \cdot \frac{1}{4}(x+1)\,dx\) | M1 | Attempt \(\int_0^2 x f(x)\,dx\) |
| \(= \left[\frac{1}{12}x^3 + \frac{1}{8}x^2\right]_0^2\) | A1 | Expression all correct |
| \(= \frac{7}{6}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| mean \(<\) median \(<\) mode \(\Rightarrow\) negative skew | M1A1 | Comparison, both |
# Question 7:
## Part (a)
| $8k = 1,\; k = \frac{1}{8}$ | B1 | cso |
## Part (b)
| $F(m) = 0.5$ | M1 | |
| $x^2 + 2x - 4 = 0$ | A1 | |
| $x = \sqrt{5} - 1 = 1.236$ | A1 | awrt 1.24 |
## Part (c)
| $f(x) = \frac{1}{4}(x+1),\quad 0 \leq x \leq 2$ | M1A1 | Differentiation, all correct |
| $f(x) = 0$, otherwise | A1 | 0 and ranges |
## Part (d)
| Graph of $f(x)$: linear from $\frac{1}{4}$ at $x=0$ to $\frac{3}{4}$ at $x=2$, zero elsewhere | B1, B1, B1 | Values & labels; slope; $f(x)=0$ |
## Part (e)
| mode $= 2$ | B1 | |
## Part (f)
| $E(X) = \int_0^2 x \cdot \frac{1}{4}(x+1)\,dx$ | M1 | Attempt $\int_0^2 x f(x)\,dx$ |
| $= \left[\frac{1}{12}x^3 + \frac{1}{8}x^2\right]_0^2$ | A1 | Expression all correct |
| $= \frac{7}{6}$ | A1 | |
## Part (g)
| mean $<$ median $<$ mode $\Rightarrow$ negative skew | M1A1 | Comparison, both |7. A continuous random variable $X$ has cumulative distribution function $\mathrm { F } ( x )$ given by
$$\mathrm { F } ( x ) = \left\{ \begin{array} { l r }
0 , & x < 0 \\
k x ^ { 2 } + 2 k x , & 0 \leq x \leq 2 \\
8 k , & x > 2
\end{array} \right.$$
\begin{enumerate}[label=(\alph*)]
\item Show that $k = \frac { 1 } { 8 }$.
\item Find the median of $X$.
\item Find the probability density function $\mathrm { f } ( x )$.
\item Sketch $\mathrm { f } ( x )$ for all values of $x$.
\item Write down the mode of $X$.
\item Find $\mathrm { E } ( X )$.
\item Comment on the skewness of this distribution.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2002 Q7 [16]}}