| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2005 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Breaking/cutting problems |
| Difficulty | Standard +0.8 This S2 question requires careful conceptual understanding that X represents the *shorter* piece (not the break point), necessitating recognition that X ranges from 0 to l with a non-uniform transformation from the uniform break point. Parts (a)-(c) test this insight, while part (d) adds independence. This is above-average difficulty due to the conceptual subtlety rather than computational complexity. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| Continuous uniform / Rectangular distribution | B1 | |
| \(f(x) = \begin{cases} \frac{1}{l}, & 0 \leq x \leq l \\ 0, & \text{otherwise} \end{cases}\) | B1, B1 | (3 marks total) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P\left(X < \frac{1}{3}l\right) = \frac{1}{l} \times \frac{l}{3} = \frac{1}{3}\) | M1, A1 | This is \(\frac{1}{l} \times \frac{l}{3}\); (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(E(X) = \frac{1}{2}l\) | B1 | (1 mark) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P\left(\text{Both} < \frac{1}{3}l\right) = \left(\frac{1}{3}\right)^2 = \frac{1}{9}\) | M1, A1 | \((b)^2\); (2 marks) |
## Question 3:
**(a)**
Continuous uniform / Rectangular distribution | B1 |
$f(x) = \begin{cases} \frac{1}{l}, & 0 \leq x \leq l \\ 0, & \text{otherwise} \end{cases}$ | B1, B1 | (3 marks total)
**(b)**
$P\left(X < \frac{1}{3}l\right) = \frac{1}{l} \times \frac{l}{3} = \frac{1}{3}$ | M1, A1 | This is $\frac{1}{l} \times \frac{l}{3}$; (2 marks)
**(c)**
$E(X) = \frac{1}{2}l$ | B1 | (1 mark)
**(d)**
$P\left(\text{Both} < \frac{1}{3}l\right) = \left(\frac{1}{3}\right)^2 = \frac{1}{9}$ | M1, A1 | $(b)^2$; (2 marks)
---3. A rod of length $2 l$ was broken into 2 parts. The point at which the rod broke is equally likely to be anywhere along the rod. The length of the shorter piece of rod is represented by the random variable $X$.
\begin{enumerate}[label=(\alph*)]
\item Write down the name of the probability density function of $X$, and specify it fully.
\item Find $\mathrm { P } \left( X < \frac { 1 } { 3 } l \right)$.
\item Write down the value of $\mathrm { E } ( X )$.
Two identical rods of length $2 l$ are broken.
\item Find the probability that both of the shorter pieces are of length less than $\frac { 1 } { 3 } l$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2005 Q3 [8]}}