| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Difficulty | Standard +0.3 This is a standard S2 piecewise pdf question requiring routine integration and cdf construction. While it has multiple parts and involves piecewise functions, all techniques are straightforward applications of textbook methods: verifying ∫f(x)dx=1, reading probabilities from the pdf, integrating to find F(x) in each domain, and using F(x) to find probabilities. The algebra is manageable and no novel insight is 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 | Guidance |
|---|---|---|
| (a) \(\int_0^{10} f(x)\,dx = \frac{3}{7} + \frac{6}{875}\int_5^{10}(10x - x^2)\,dx = \frac{3}{7} + \frac{6}{875}\left[5x^2 - \frac{1}{3}x^3\right]_5^{10}\) | M1, A1, A1 | |
| \(= \frac{3}{7} + \frac{6}{875}\left[500 - \frac{1000}{3} - 125 + \frac{3}{3}\right] = \frac{3}{7} + \frac{4}{7} = 1\) | M1, A1 | |
| Also \(f(x) \geq 0\) for all \(x\), so \(f\) is a p.d.f. | A1 | |
| (b) \(P(X < 1) = \frac{1}{2} \times 1 \times \frac{3}{175} = \frac{3}{175}\) | M1, A1 | |
| (c) \(F(x) = 0\) (\(x < 0\)) | B1, M1, A1 | |
| \(F(x) = \frac{3}{175}x^2\) (\(0 \leq x \leq 5\)) | M1, A1 | |
| \(F(x) = \frac{3}{7} + \int_5^x \frac{6}{875}(10u - u^2)\,du = \frac{30x^2 - 2x^3 - 125}{875}\) (\(5 < x < 10\)) | M1, A1 | |
| \(F(x) = 1\) (\(x > 10\)) | B1 | |
| (d) \(F(7) - F(2) = \frac{899}{875} - \frac{12}{175} = \frac{899}{875}\) | M1, A1 | 17 marks total |
(a) $\int_0^{10} f(x)\,dx = \frac{3}{7} + \frac{6}{875}\int_5^{10}(10x - x^2)\,dx = \frac{3}{7} + \frac{6}{875}\left[5x^2 - \frac{1}{3}x^3\right]_5^{10}$ | M1, A1, A1 |
$= \frac{3}{7} + \frac{6}{875}\left[500 - \frac{1000}{3} - 125 + \frac{3}{3}\right] = \frac{3}{7} + \frac{4}{7} = 1$ | M1, A1 |
Also $f(x) \geq 0$ for all $x$, so $f$ is a p.d.f. | A1 |
(b) $P(X < 1) = \frac{1}{2} \times 1 \times \frac{3}{175} = \frac{3}{175}$ | M1, A1 |
(c) $F(x) = 0$ ($x < 0$) | B1, M1, A1 |
$F(x) = \frac{3}{175}x^2$ ($0 \leq x \leq 5$) | M1, A1 |
$F(x) = \frac{3}{7} + \int_5^x \frac{6}{875}(10u - u^2)\,du = \frac{30x^2 - 2x^3 - 125}{875}$ ($5 < x < 10$) | M1, A1 |
$F(x) = 1$ ($x > 10$) | B1 |
(d) $F(7) - F(2) = \frac{899}{875} - \frac{12}{175} = \frac{899}{875}$ | M1, A1 | 17 marks total |A continuous random variable $X$ has the probability density function
$$\text{f}(x) = \frac{6x}{175} \quad 0 \leq x < 5,$$
$$\text{f}(x) = \frac{6x(10-x)}{875} \quad 5 \leq x \leq 10,$$
$$\text{f}(x) = 0 \quad \text{otherwise}.$$
\begin{enumerate}[label=(\alph*)]
\item Verify that f is a probability density function. [6 marks]
\item Write down the probability that $X < 1$. [2 marks]
\item Find the cumulative distribution function of $X$, carefully showing how it changes for different domains. [7 marks]
\item Find the probability that $2 < X < 7$. [2 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q7 [17]}}