| Exam Board | Edexcel |
|---|---|
| Module | FS2 AS (Further Statistics 2 AS) |
| Year | 2022 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Find or specify CDF |
| Difficulty | Standard +0.3 This is a straightforward Further Statistics 2 question requiring students to read a PDF from a graph, calculate a probability using area under the curve, and write down the CDF formula by integrating the PDF. While it's Further Maths content, these are standard textbook exercises requiring only basic integration and understanding of PDF/CDF relationships, making it slightly easier than average overall. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03e Find cdf: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(P(X<4) = \frac{(4-1)\times 0.1}{2}\) or \(\int_1^4 \frac{1}{30}(x-1)\,dx\) | M1 | Use of area of triangle or integration with limits; condone \(\frac{4\times0.1}{2}\) or \(\int_1^4 \frac{1}{30}(x)\,dx\) for M1 |
| \(= 0.15\) | A1 | 0.15 oe |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(\frac{(7-1)\times0.2}{2}+\ldots\) or \(\int_1^7 \frac{1}{30}(x-1)\,dx+\ldots\) or \(\int 0.1\,dx\) | M1 | Complete method for finding cdf including area from \(1\leq x<7\); condone one slip |
| \(\ldots 0.1(x-7)\) or \(\ldots\int_7^x 0.1\,dt\) or \(F(7)=0.1x+c\) or \(F(11)=0.1x+c\) | M1 | Attempt at area from \(7\leq x\leq 11\) |
| \(F(x)=0.1x-0.1\) \(\left[\text{for } 7\leq x\leq 11\right]\) | A1 | \(0.1x-0.1\) |
## Question 2:
### Part (a):
| Working | Mark | Guidance |
|---------|------|----------|
| $P(X<4) = \frac{(4-1)\times 0.1}{2}$ or $\int_1^4 \frac{1}{30}(x-1)\,dx$ | M1 | Use of area of triangle or integration with limits; condone $\frac{4\times0.1}{2}$ or $\int_1^4 \frac{1}{30}(x)\,dx$ for M1 |
| $= 0.15$ | A1 | 0.15 oe |
**(2 marks)**
### Part (b):
| Working | Mark | Guidance |
|---------|------|----------|
| $\frac{(7-1)\times0.2}{2}+\ldots$ or $\int_1^7 \frac{1}{30}(x-1)\,dx+\ldots$ or $\int 0.1\,dx$ | M1 | Complete method for finding cdf including area from $1\leq x<7$; condone one slip |
| $\ldots 0.1(x-7)$ or $\ldots\int_7^x 0.1\,dt$ or $F(7)=0.1x+c$ or $F(11)=0.1x+c$ | M1 | Attempt at area from $7\leq x\leq 11$ |
| $F(x)=0.1x-0.1$ $\left[\text{for } 7\leq x\leq 11\right]$ | A1 | $0.1x-0.1$ |
**(3 marks)**
---\begin{enumerate}
\item The graph shows the probability density function $\mathrm { f } ( x )$ of the continuous random variable $X$\\
\includegraphics[max width=\textwidth, alt={}, center]{128c408d-3e08-4f74-8f19-d33ecd5c882f-04_951_1365_322_331}\\
(a) Find $\mathrm { P } ( X < 4 )$\\
(b) Specify the cumulative distribution function of $X$ for $7 \leqslant x \leqslant 11$
\end{enumerate}
\hfill \mbox{\textit{Edexcel FS2 AS 2022 Q2 [5]}}