| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Cumulative distribution functions |
| Type | CDF to PDF derivation |
| Difficulty | Moderate -0.3 This is a straightforward S2 question testing basic understanding of CDFs and PDFs. Part (a) requires simple substitution into the given CDF formula. Part (b) involves differentiating the CDF to find the PDF, which is a standard procedure. Part (c) asks for a sketch of a linear function. All parts are routine applications of core concepts with no problem-solving insight required, 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 |
|---|---|
| \(= 1 - F(5) = 1 - \frac{1}{64}(80 - 25) = \frac{9}{64}\) | M1 A1 |
| Answer | Marks |
|---|---|
| \(f(x) = F'(x) = \frac{1}{64}(16 - 2x)\) | M1 A1 |
| \(\therefore f(x) = \begin{cases} \frac{1}{32}(8 - x), & 0 \le x \le 8, \\ 0, & \text{otherwise} \end{cases}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Graph showing \(f(x)\) with y-intercept \(\frac{1}{4}\) and x-intercept at \(x = 8\) | B3 | (8) |
**Part (a)**
$= 1 - F(5) = 1 - \frac{1}{64}(80 - 25) = \frac{9}{64}$ | M1 A1 |
**Part (b)**
$f(x) = F'(x) = \frac{1}{64}(16 - 2x)$ | M1 A1 |
$\therefore f(x) = \begin{cases} \frac{1}{32}(8 - x), & 0 \le x \le 8, \\ 0, & \text{otherwise} \end{cases}$ | A1 |
**Part (c)**
Graph showing $f(x)$ with y-intercept $\frac{1}{4}$ and x-intercept at $x = 8$ | B3 | (8)
---The continuous random variable $X$ has the following cumulative distribution function:
$$F(x) = \begin{cases}
0, & x < 0, \\
\frac{1}{64}(16x - x^2), & 0 \leq x \leq 8, \\
1, & x > 8.
\end{cases}$$
\begin{enumerate}[label=(\alph*)]
\item Find $P(X > 5)$. [2 marks]
\item Find and specify fully the probability density function $f(x)$ of $X$. [3 marks]
\item Sketch $f(x)$ for all values of $x$. [3 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q2 [8]}}