| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 13 |
| 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 probability density function question requiring standard techniques: finding k by integration (∫k(10-t)dt = 1), computing E(T) = ∫tf(t)dt, and solving ∫f(t)dt = 0.95 for the percentile. The linear pdf makes all integrations simple, and part (d) requires only basic interpretation. Slightly easier than average due to the simple functional form and routine application of standard methods. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Graph: straight line from \((0, 10k)\) to \((10, 0)\); on x-axis otherwise | B2 | |
| \(\frac{1}{2} \times 10 \times 10k = 1\) \(k = \frac{1}{50}\) | M1 A1 | |
| (b) \(E(T) = \int_0^{10} f(t) \, dt = \frac{1}{50} \int_0^{10} 10t - t^2 \, dt = \frac{1}{50}\left[5t^2 - \frac{t^3}{3}\right]_0^{10} = 3\frac{1}{3}\) | M1 A1 M1 A1 | |
| (c) From graph, \(\frac{1}{2}(10-p)\left(\frac{10-p}{50}\right) = \frac{5}{100}\) \((10-p)^2 = 5\) \(p = 7.76\) | M1 A1 M1 A1 | |
| (d) Some wait more than 10 mins; more gradual slope needed | B1 B1 | 13 marks total |
(a) Graph: straight line from $(0, 10k)$ to $(10, 0)$; on x-axis otherwise | B2 |
$\frac{1}{2} \times 10 \times 10k = 1$ $k = \frac{1}{50}$ | M1 A1 |
(b) $E(T) = \int_0^{10} f(t) \, dt = \frac{1}{50} \int_0^{10} 10t - t^2 \, dt = \frac{1}{50}\left[5t^2 - \frac{t^3}{3}\right]_0^{10} = 3\frac{1}{3}$ | M1 A1 M1 A1 |
(c) From graph, $\frac{1}{2}(10-p)\left(\frac{10-p}{50}\right) = \frac{5}{100}$ $(10-p)^2 = 5$ $p = 7.76$ | M1 A1 M1 A1 |
(d) Some wait more than 10 mins; more gradual slope needed | B1 B1 | 13 marks totalThe waiting time, in minutes, at a dentist is modelled by the continuous random variable $T$ with probability density function
$$f(t) = k(10 - t) \quad 0 \leq t \leq 10,$$
$$f(t) = 0 \quad \text{otherwise}.$$
\begin{enumerate}[label=(\alph*)]
\item Sketch the graph of $f(t)$ and find the value of $k$. [4 marks]
\item Find the mean value of $T$. [4 marks]
\item Find the 95th percentile of $T$. [3 marks]
\item State whether you consider this function to be a sensible model for $T$ and suggest how it could be modified to provide a better model. [2 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q4 [13]}}