| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2002 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Waiting time applications |
| Difficulty | Moderate -0.8 This is a straightforward application of continuous uniform distribution requiring only basic definitions and calculations. Part (a) asks for model identification (uniform on [0,14]), parts (b-d) involve standard formulas for mean, CDF, and probability calculation with no problem-solving insight needed—purely routine S2 material. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03c Calculate mean/variance: by integration5.03e Find cdf: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| Continuous Uniform (Rectangular), \(X \sim U[0,14]\) | B1, B1 | Both parameters correct |
| Answer | Marks | Guidance |
|---|---|---|
| \(E(X) = \frac{14+0}{2} = 7\) | M1A1 | Form & substitute, answer 7 |
| Mean arrival time is 8.02am | A1 | 8.02am |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X \leq x) = \int_0^x \frac{1}{14}\,dt = \frac{x}{14}\) | M1, A1 | Integral, \(\frac{x}{14}\) |
| \(F(x) = 0\), \(x < 0\) | ||
| \(F(x) = \frac{x}{14}\), \(0 \leq x \leq 14\) | B1ft | Centre |
| \(F(x) = 1\), \(x > 14\) | Ends |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X > 10) = 1 - F(10)\) | B1 | Require '1 minus' or valid integral |
| \(= 1 - \frac{10}{14} = \frac{2}{7}\) | M1, A1 | \(\frac{2}{7}\) |
# Question 4:
## Part (a)
| Continuous Uniform (Rectangular), $X \sim U[0,14]$ | B1, B1 | Both parameters correct |
## Part (b)
| $E(X) = \frac{14+0}{2} = 7$ | M1A1 | Form & substitute, answer 7 |
| Mean arrival time is 8.02am | A1 | 8.02am |
## Part (c)
| $P(X \leq x) = \int_0^x \frac{1}{14}\,dt = \frac{x}{14}$ | M1, A1 | Integral, $\frac{x}{14}$ |
| $F(x) = 0$, $x < 0$ | | |
| $F(x) = \frac{x}{14}$, $0 \leq x \leq 14$ | B1ft | Centre |
| $F(x) = 1$, $x > 14$ | | Ends |
## Part (d)
| $P(X > 10) = 1 - F(10)$ | B1 | Require '1 minus' or valid integral |
| $= 1 - \frac{10}{14} = \frac{2}{7}$ | M1, A1 | $\frac{2}{7}$ |
---4. Jean catches a bus to work every morning. According to the timetable the bus is due at 8 a.m., but Jean knows that the bus can arrive at a random time between five minutes early and 9 minutes late. The random variable $X$ represents the time, in minutes, after 7.55 a.m. when the bus arrives.
\begin{enumerate}[label=(\alph*)]
\item Suggest a suitable model for the distribution of $X$ and specify it fully.
\item Calculate the mean time of arrival of the bus.
\item Find the cumulative distribution function of $X$.
Jean will be late for work if the bus arrives after 8.05 a.m.
\item Find the probability that Jean is late for work.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2002 Q4 [11]}}