| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2010 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Cumulative distribution function |
| Difficulty | Moderate -0.8 This is a straightforward S2 question testing basic understanding of CDFs and uniform distributions. All parts require direct application of standard definitions: reading values from the CDF, differentiating to find the PDF, recognizing a uniform distribution, and applying standard formulas for mean and variance. No problem-solving or novel insight required—purely procedural recall. |
| Spec | 5.02e Discrete uniform distribution5.03a Continuous random variables: pdf and cdf5.03d E(g(X)): general expectation formula5.03e Find cdf: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(P(X < 0) = F(0) = \frac{2}{6} = \frac{1}{3}\) | M1 A1 | (2 marks) |
| (b) \(f(x) = \frac{dF(x)}{dx}\) where \(f(x) = \begin{cases} \frac{1}{6} & -2 \leq x \leq 4 \\ 0 & \text{otherwise} \end{cases}\) | M1 A1 B1 | (3 marks) |
| (c) Continuous Uniform (Rectangular) distribution | B1 | (1 mark) |
| (d) Mean \(= 1\) | B1 | Variance \(= \frac{(4-(-2))^2}{12} = 3\) |
| (e) \(P(X = 1) = 0\) | B1 | (1 mark) |
| Total [10] |
**(a)** $P(X < 0) = F(0) = \frac{2}{6} = \frac{1}{3}$ | M1 A1 | (2 marks)
**(b)** $f(x) = \frac{dF(x)}{dx}$ where $f(x) = \begin{cases} \frac{1}{6} & -2 \leq x \leq 4 \\ 0 & \text{otherwise} \end{cases}$ | M1 A1 B1 | (3 marks)
**(c)** Continuous Uniform (Rectangular) distribution | B1 | (1 mark)
**(d)** Mean $= 1$ | B1 | Variance $= \frac{(4-(-2))^2}{12} = 3$ | M1 A1 | (3 marks)
**(e)** $P(X = 1) = 0$ | B1 | (1 mark)
| **Total [10]**
---2. A continuous random variable $X$ has cumulative distribution function
$$\mathrm { F } ( x ) = \begin{cases} 0 , & x < - 2 \\ \frac { x + 2 } { 6 } , & - 2 \leqslant x \leqslant 4 \\ 1 , & x > 4 \end{cases}$$
\begin{enumerate}[label=(\alph*)]
\item Find $\mathrm { P } ( X < 0 )$.
\item Find the probability density function $\mathrm { f } ( x )$ of $X$.
\item Write down the name of the distribution of $X$.
\item Find the mean and the variance of $X$.
\item Write down the value of $\mathrm { P } ( X = 1 )$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2010 Q2 [10]}}