| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Cumulative distribution functions |
| Type | Find quantiles from CDF |
| Difficulty | Standard +0.3 This is a straightforward S2 question requiring standard techniques: solving F(x) = 0.5 and 0.25/0.75 for quantiles, differentiating the CDF to find the PDF, and sketching a linear function. All steps are routine applications of definitions with no conceptual challenges, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03e Find cdf: by integration5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Need \(F(x) = 0.5\), so \(x^2 = 58\); \(x = \sqrt{58} = 7.62\) | M1 A1 | |
| (b) \(\frac{1}{84}(p^2 - 16) = 0.25\); \(p = 6.083\); \(\frac{1}{84}(q^2 - 16) = 0.75\); \(q = 8.888\) | M1 A1 A1 | |
| \(\text{IQR} = 8.888 - 6.083 = 2.81\) | A1 | |
| (c) \(f(x) = F'(x) = \frac{x}{42}\), \(4 \leq x \leq 10\); \(f(x) = 0\) otherwise | M1 A1 A1 | |
| (d) Graph drawn; Mode \(= 10\); maximum value of \(f(x)\) on graph | B1 M1 A1 | Total: 12 marks |
(a) Need $F(x) = 0.5$, so $x^2 = 58$; $x = \sqrt{58} = 7.62$ | M1 A1 |
(b) $\frac{1}{84}(p^2 - 16) = 0.25$; $p = 6.083$; $\frac{1}{84}(q^2 - 16) = 0.75$; $q = 8.888$ | M1 A1 A1 |
$\text{IQR} = 8.888 - 6.083 = 2.81$ | A1 |
(c) $f(x) = F'(x) = \frac{x}{42}$, $4 \leq x \leq 10$; $f(x) = 0$ otherwise | M1 A1 A1 |
(d) Graph drawn; Mode $= 10$; maximum value of $f(x)$ on graph | B1 M1 A1 | Total: 12 marks4. A continuous random variable $X$ has the cumulative distribution function
$$\mathrm { F } ( x ) = \left\{ \begin{array} { l }
0 \\
\frac { 1 } { 84 } \left( x ^ { 2 } - 16 \right) \\
1
\end{array} \right.$$
$$\begin{aligned}
& x < 4 , \\
& 4 \leq x \leq 10 , \\
& x > 10 .
\end{aligned}$$
\begin{enumerate}[label=(\alph*)]
\item Find the median value of $X$.
\item Find the interquartile range for $X$.
\item Find the probability density function $\mathrm { f } ( x )$ of $X$.
\item Sketch the graph of $\mathrm { f } ( x )$ and hence write down the mode of $X$, explaining how you obtain your answer from the graph.
\section*{STATISTICS 2 (A) TEST PAPER 1 Page 2}
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q4 [12]}}