| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2016 |
| Session | October |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | PDF with multiple constants |
| Difficulty | Standard +0.3 This is a standard S2 PDF question requiring routine application of fundamental properties: using ∫f(x)dx=1 to find constants, calculating E(X) by integration, and finding quartiles. All techniques are textbook exercises with straightforward integration of linear and constant functions. Slightly easier than average due to the simple functional forms involved. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{5\times k}{2}+(10.5-5)\times k=1\); \(8k=1\); \(k=\frac{1}{8}\) | M1, A1cso | Using sum of triangle and rectangle areas \(=1\) or integration, and \(5m=k\); correct solution or finding \(m\) and verifying \(k=\frac{1}{8}\) |
| Answer | Marks |
|---|---|
| \(m=\frac{1}{40}\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(E(X)=\int_0^5 \frac{1}{40}x^2\,dx+\int_5^{10.5}\frac{1}{8}x\,dx\); \(=\left[\frac{1}{40}\left(\frac{x^3}{3}\right)\right]_0^5+\left[\frac{1}{8}\left(\frac{x^2}{2}\right)\right]_5^{10.5}=\frac{1223}{192}\) | M1, A1ft A1 | \(xf(x)\) for both parts and attempt to integrate; correct integration with limits; awrt 6.37 |
| Answer | Marks | Guidance |
|---|---|---|
| LQ: \(\frac{x\times\frac{1}{40}\times x}{2}=0.25\) or \(\int_0^{lq}\frac{1}{40}x\,dx=0.25\); UQ: \(\frac{5\times\frac{1}{8}}{2}+(uq-5)\times\frac{1}{8}=0.75\); \(LQ=\sqrt{20}\) (awrt 4.47), \(UQ=8.5\); \(IQR=8.5-\sqrt{20}=4.02786\ldots\) | M1, M1, A1, A1 | 1st M1: correct method for LQ; 2nd M1: correct method for UQ; A1 for either LQ awrt 4.47 or UQ \(=8.5\); A1 for IQR awrt 4.03 |
# Question 4:
## Part (a)(i)
| $\frac{5\times k}{2}+(10.5-5)\times k=1$; $8k=1$; $k=\frac{1}{8}$ | M1, A1cso | Using sum of triangle and rectangle areas $=1$ or integration, and $5m=k$; correct solution or finding $m$ and verifying $k=\frac{1}{8}$ |
## Part (a)(ii)
| $m=\frac{1}{40}$ | B1 | |
## Part (b)
| $E(X)=\int_0^5 \frac{1}{40}x^2\,dx+\int_5^{10.5}\frac{1}{8}x\,dx$; $=\left[\frac{1}{40}\left(\frac{x^3}{3}\right)\right]_0^5+\left[\frac{1}{8}\left(\frac{x^2}{2}\right)\right]_5^{10.5}=\frac{1223}{192}$ | M1, A1ft A1 | $xf(x)$ for both parts and attempt to integrate; correct integration with limits; awrt **6.37** |
## Part (c)
| LQ: $\frac{x\times\frac{1}{40}\times x}{2}=0.25$ or $\int_0^{lq}\frac{1}{40}x\,dx=0.25$; UQ: $\frac{5\times\frac{1}{8}}{2}+(uq-5)\times\frac{1}{8}=0.75$; $LQ=\sqrt{20}$ (awrt 4.47), $UQ=8.5$; $IQR=8.5-\sqrt{20}=4.02786\ldots$ | M1, M1, A1, A1 | 1st M1: correct method for LQ; 2nd M1: correct method for UQ; A1 for either LQ awrt 4.47 or UQ $=8.5$; A1 for IQR awrt **4.03** |
---4.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{4ecee051-3a6f-4c12-8c53-926e8c3e241f-14_451_976_233_484}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
A continuous random variable $X$ has the probability density function $\mathrm { f } ( x )$ shown in Figure 1
$$\mathrm { f } ( x ) = \begin{cases} m x & 0 \leqslant x \leqslant 5 \\ k & 5 < x \leqslant 10.5 \\ 0 & \text { otherwise } \end{cases}$$
where $m$ and $k$ are constants.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Show that $k = \frac { 1 } { 8 }$
\item Find the value of $m$
\end{enumerate}\item Find $\mathrm { E } ( X )$
\item Find the interquartile range of $X$
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2016 Q4 [10]}}