| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2023 |
| Session | October |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Waiting time applications |
| Difficulty | Moderate -0.8 This is a straightforward S2 uniform distribution question requiring basic understanding of the setup (showing α and β are given values), writing down the standard PDF formula, and applying memorized formulas for E(T) and Var(T). The context requires minimal problem-solving—students just need to add fixed times (3 + up to 15 + 29) and apply textbook formulas. Easier than average A-level. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(3 + [0] + 29 = 32\) | B1* | For \(3+[0]+29\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(3 + 15 + 29 = 47\) | B1* | For \(3+15+29\). Allow \(32+15\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(f(t) = \begin{cases}\frac{1}{15} & 32 \leq t \leq 47 \\ 0 & \text{otherwise}\end{cases}\) | M1 | For \(f(t)=\frac{1}{15}\), \(32\leq t\leq 47\). Allow use of \(<\) instead of \(\leq\). Must have \(f(t)\) and an inequality |
| A1 | Fully correct pdf. Must be \(f(t)\) and \(t\). Allow use of \(<\). Allow equivalent for 0 otherwise |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \([E(T)=]\ 39.5\) oe | B1 | For 39.5 oe |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \([\text{Var}(T)=]\frac{(47-32)^2}{12}\) | M1 | For use of \(\text{Var}(T)=\frac{(\beta-\alpha)^2}{12}\) |
| \(\frac{75}{4} = 18.75\) | A1 | For 18.75 oe |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \((40-32)\times\frac{1}{15}\) | M1 | For use of \((40-\alpha)\times\frac{1}{\beta-\alpha}\) |
| \(= \frac{8}{15}\) | A1 | For \(\frac{8}{15}\) oe. Allow awrt 0.533 |
# Question 3:
## Part (a)(i)
| Answer | Mark | Guidance |
|--------|------|----------|
| $3 + [0] + 29 = 32$ | B1* | For $3+[0]+29$ |
## Part (a)(ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $3 + 15 + 29 = 47$ | B1* | For $3+15+29$. Allow $32+15$ |
## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $f(t) = \begin{cases}\frac{1}{15} & 32 \leq t \leq 47 \\ 0 & \text{otherwise}\end{cases}$ | M1 | For $f(t)=\frac{1}{15}$, $32\leq t\leq 47$. Allow use of $<$ instead of $\leq$. Must have $f(t)$ and an inequality |
| | A1 | Fully correct pdf. Must be $f(t)$ and $t$. Allow use of $<$. Allow equivalent for 0 otherwise |
## Part (c)(i)
| Answer | Mark | Guidance |
|--------|------|----------|
| $[E(T)=]\ 39.5$ oe | B1 | For 39.5 oe |
## Part (c)(ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $[\text{Var}(T)=]\frac{(47-32)^2}{12}$ | M1 | For use of $\text{Var}(T)=\frac{(\beta-\alpha)^2}{12}$ |
| $\frac{75}{4} = 18.75$ | A1 | For 18.75 oe |
## Part (d)
| Answer | Mark | Guidance |
|--------|------|----------|
| $(40-32)\times\frac{1}{15}$ | M1 | For use of $(40-\alpha)\times\frac{1}{\beta-\alpha}$ |
| $= \frac{8}{15}$ | A1 | For $\frac{8}{15}$ oe. Allow awrt 0.533 |
---\begin{enumerate}
\item Every morning Navtej travels from home to work. Navtej leaves home at a random time between 08:00 and 08:15
\end{enumerate}
\begin{itemize}
\item It always takes Navtej 3 minutes to walk to the bus stop
\item Buses run every 15 minutes and Navtej catches the first bus that arrives
\item Once Navtej has caught the bus it always takes a further 29 minutes for Navtej to reach work
\end{itemize}
The total time, $T$ minutes, for Navtej's journey from home to work is modelled by a continuous uniform distribution over the interval $[ \alpha , \beta ]$\\
(a) (i) Show that $\alpha = 32$\\
(ii) Show that $\beta = 47$\\
(b) State fully the probability density function for this distribution.\\
(c) Find the value of\\
(i) $\mathrm { E } ( T )$\\
(ii) $\operatorname { Var } ( T )$\\
(d) Find the probability that the time for Navtej's journey is within 5 minutes of 35 minutes.
\hfill \mbox{\textit{Edexcel S2 2023 Q3 [9]}}