| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| 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 continuous uniform distribution question requiring only recognition of the distribution name, routine integration to find the CDF, a simple probability calculation, and qualitative reasoning about model limitations. All parts are standard textbook exercises with no problem-solving insight required, making it easier than average for A-level. |
| Spec | 5.02e Discrete uniform distribution5.03a Continuous random variables: pdf and cdf5.03e Find cdf: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| (a) continuous uniform | B1 | |
| (b) \(F(t) = \int_{-4}^{t} \frac{1}{8} \, dx = \frac{1}{8}[x]_{-4}^{t} = \frac{1}{8}(t + 4)\); \(F(x) = \begin{cases} 0, & x < -4, \\ \frac{1}{8}(x + 4), & -4 \leq x \leq 4, \\ 1, & x > 4. \end{cases}\) | M1, M1, A1, A1 | |
| (c) \(= P(-1.5 \leq s \leq 1.5) = 3 \times \frac{1}{8} = \frac{3}{8}\) | M1, M1, A1 | |
| (d) e.g. gives zero prob. of more than 4 cm error and doesn't suggest higher prob. density near 0 as would be likely | B2 | (10 marks) |
**(a)** continuous uniform | B1 |
**(b)** $F(t) = \int_{-4}^{t} \frac{1}{8} \, dx = \frac{1}{8}[x]_{-4}^{t} = \frac{1}{8}(t + 4)$; $F(x) = \begin{cases} 0, & x < -4, \\ \frac{1}{8}(x + 4), & -4 \leq x \leq 4, \\ 1, & x > 4. \end{cases}$ | M1, M1, A1, A1 |
**(c)** $= P(-1.5 \leq s \leq 1.5) = 3 \times \frac{1}{8} = \frac{3}{8}$ | M1, M1, A1 |
**(d)** e.g. gives zero prob. of more than 4 cm error and doesn't suggest higher prob. density near 0 as would be likely | B2 | (10 marks)A class of children are each asked to draw a line that they think is 10 cm long without using a ruler. The teacher models how many centimetres each child's line is longer than 10 cm by the random variable $X$ and believes that $X$ has the following probability density function:
$$f(x) = \begin{cases}
\frac{1}{8}, & -4 \leq x \leq 4, \\
0, & \text{otherwise}.
\end{cases}$$
\begin{enumerate}[label=(\alph*)]
\item Write down the name of this distribution. [1]
\item Define fully the cumulative distribution function F(x) of $X$. [4]
\item Calculate the proportion of children making an error of less than 15\% according to this model. [3]
\item Give two reasons why this may not be a very suitable model. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q3 [10]}}