| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2007 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Find parameters from given statistics |
| Difficulty | Moderate -0.3 This is a straightforward S2 question testing standard uniform distribution formulas. Part (b) requires solving two simultaneous equations using basic properties, while parts (c)-(e) apply textbook formulas to a simple context (X uniform on [0,150]). All steps are routine with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)5.03a Continuous random variables: pdf and cdf5.03d E(g(X)): general expectation formula |
| Answer | Marks | Guidance |
|---|---|---|
| \(f(x) = \begin{cases} \frac{1}{\beta - \alpha}, & \alpha < x < \beta, \\ 0, & \text{otherwise.} \end{cases}\) | function including inequality, 0 otherwise | B1, B1 (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{\alpha + \beta}{2} = 2\), \(\frac{3 - \alpha}{\beta - \alpha} = \frac{5}{8}\) | or equivalent | B1, B1 |
| \(\alpha + \beta = 4\) \(3\alpha + 5\beta = 24\) | ||
| \(3(4 - \beta) + 5\beta = 24\) \(2\beta = 12\) \(\beta = 6\) | attempt to solve 2 eqns | M1 |
| \(\alpha = -2\) | both | A1 (4 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(E(X) = \frac{150 + 0}{2} = 75\) cm | 75 | B1 (1 mark) |
| Answer | Marks | Guidance |
|---|---|---|
| Standard deviation \(= \sqrt{\frac{1}{12}(150 - 0)^2}\) | M1 | |
| \(= 43.30127\ldots\) cm | \(25\sqrt{3}\) or awrt 43.3 | A1 (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X < 30) + P(X > 120) = \frac{30}{150} + \frac{30}{150}\) | 1st or at least one fraction, + or double | M1, M1 |
| \(= \frac{60}{150}\) or \(\frac{2}{5}\) or 0.4 or equivalent fraction | A1 (3 marks) |
**Part (a)**
$f(x) = \begin{cases} \frac{1}{\beta - \alpha}, & \alpha < x < \beta, \\ 0, & \text{otherwise.} \end{cases}$ | function including inequality, 0 otherwise | B1, B1 (2 marks) |
**Part (b)**
$\frac{\alpha + \beta}{2} = 2$, $\frac{3 - \alpha}{\beta - \alpha} = \frac{5}{8}$ | or equivalent | B1, B1 |
$\alpha + \beta = 4$ $3\alpha + 5\beta = 24$ | | |
$3(4 - \beta) + 5\beta = 24$ $2\beta = 12$ $\beta = 6$ | attempt to solve 2 eqns | M1 |
$\alpha = -2$ | both | A1 (4 marks) |
**Part (c)**
$E(X) = \frac{150 + 0}{2} = 75$ cm | 75 | B1 (1 mark) |
**Part (d)**
Standard deviation $= \sqrt{\frac{1}{12}(150 - 0)^2}$ | | M1 |
$= 43.30127\ldots$ cm | $25\sqrt{3}$ or awrt 43.3 | A1 (2 marks) |
**Part (e)**
$P(X < 30) + P(X > 120) = \frac{30}{150} + \frac{30}{150}$ | 1st or at least one fraction, + or double | M1, M1 |
$= \frac{60}{150}$ or $\frac{2}{5}$ or 0.4 or equivalent fraction | | A1 (3 marks) |
---5. The continuous random variable $X$ is uniformly distributed over the interval $\alpha < x < \beta$.
\begin{enumerate}[label=(\alph*)]
\item Write down the probability density function of $X$, for all $x$.
\item Given that $\mathrm { E } ( X ) = 2$ and $\mathrm { P } ( X < 3 ) = \frac { 5 } { 8 }$ find the value of $\alpha$ and the value of $\beta$.
A gardener has wire cutters and a piece of wire 150 cm long which has a ring attached at one end. The gardener cuts the wire, at a randomly chosen point, into 2 pieces. The length, in cm, of the piece of wire with the ring on it is represented by the random variable $X$. Find
\item $\mathrm { E } ( X )$,
\item the standard deviation of $X$,
\item the probability that the shorter piece of wire is at most 30 cm long.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2007 Q5 [12]}}