| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Standard applied PDF calculations |
| Difficulty | Standard +0.3 This is a straightforward S2 continuous distribution question requiring standard techniques: sketching a quadratic pdf, identifying the mean from symmetry, calculating probabilities via integration, and applying conditional probability. The integrations are routine (polynomial), and the conditional probability setup is standard. While multi-part with 11 marks total, each component is textbook-level with no novel insight required, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf |
| Answer | Marks |
|---|---|
| (a) Graph drawn through \((0, 0)\) and \((3, 0)\) | B2; B1 |
| (b) 150 hours | |
| (c) \(P(X < 2) = \int_0^2 f(x)\, dx = \frac{2}{3}\left[\frac{3x^2}{2} - \frac{x^3}{3}\right]_0^2 = \frac{2}{3}\left(6 - \frac{8}{3}\right) = 0.741\) | M1 A1 |
| \(P(X > 2.5) = P(X < 0.5) = \frac{2}{3}\left[\frac{3x^2}{2} - \frac{x^3}{3}\right]_0^{0.5} = \frac{2}{3}\left(\frac{3}{8} - \frac{1}{24}\right) = 0.0741\) | M1 A1 A1 |
| (d) \(0.0741 + (1 - 0.0741) = 0.286\) | M1 A1 |
| (e) Too rigid a cut-off; some bulbs might last longer than 300 hours | B1 |
| 11 |
(a) Graph drawn through $(0, 0)$ and $(3, 0)$ | B2; B1 |
(b) 150 hours | |
(c) $P(X < 2) = \int_0^2 f(x)\, dx = \frac{2}{3}\left[\frac{3x^2}{2} - \frac{x^3}{3}\right]_0^2 = \frac{2}{3}\left(6 - \frac{8}{3}\right) = 0.741$ | M1 A1 |
$P(X > 2.5) = P(X < 0.5) = \frac{2}{3}\left[\frac{3x^2}{2} - \frac{x^3}{3}\right]_0^{0.5} = \frac{2}{3}\left(\frac{3}{8} - \frac{1}{24}\right) = 0.0741$ | M1 A1 A1 |
(d) $0.0741 + (1 - 0.0741) = 0.286$ | M1 A1 |
(e) Too rigid a cut-off; some bulbs might last longer than 300 hours | B1 |
| | 11 |Light bulbs produced in a certain factory have lifetimes, in 100s of hours, whose distribution is modelled by the random variable $X$ with probability density function
$$f(x) = \frac{2x(3-x)}{9}, \quad 0 \leq x \leq 3;$$
$$f(x) = 0 \quad \text{otherwise}.$$
\begin{enumerate}[label=(\alph*)]
\item Sketch $f(x)$. [2 marks]
\item Write down the mean lifetime of a bulb. [1 mark]
\item Show that ten times as many bulbs fail before 200 hours as survive beyond 250 hours. [5 marks]
\item Given that a bulb lasts for 200 hours, find the probability that it will then last for at least another 50 hours. [2 marks]
\item State, with a reason, whether you consider that the density function $f$ is a realistic model for the lifetimes of light bulbs. [1 mark]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q4 [11]}}