| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Compare uniform with other distributions |
| Difficulty | Standard +0.3 This is a straightforward continuous uniform distribution question requiring standard integration techniques. Part (a) involves basic calculus to derive mean and variance from first principles (routine for S2), while part (b) requires calculating a probability and comparing with normal distribution using standard results. The question is slightly easier than average as it follows a predictable template with clear steps and no novel insight required. |
| Spec | 5.02e Discrete uniform distribution5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(f(x) = \frac{1}{2a}\), \(a < x \leq 3a\) | B1 M1 A1 A1 | |
| \(E(X) = \int_a^{3a} \frac{x}{2a} \, dx = \left[\frac{x^2}{4a}\right]_a^{3a} = \frac{8a^2}{4a} = 2a\) | M1 A1 A1 | |
| \(E(X^2) = \int_a^{3a} \frac{x^2}{2a} \, dx = \left[\frac{x^3}{6a}\right]_a^{3a} = \frac{13a^2}{3}\) | M1 A1 A1 | |
| \(\text{Var}(X) = \frac{4a^2}{3}\) | M1 A1 A1 | |
| (b) \(P( | X - \mu | < \sigma) = P( |
| Normal: \(P( | X - \mu | < \sigma) = P( |
(a) $f(x) = \frac{1}{2a}$, $a < x \leq 3a$ | B1 M1 A1 A1 |
$E(X) = \int_a^{3a} \frac{x}{2a} \, dx = \left[\frac{x^2}{4a}\right]_a^{3a} = \frac{8a^2}{4a} = 2a$ | M1 A1 A1 |
$E(X^2) = \int_a^{3a} \frac{x^2}{2a} \, dx = \left[\frac{x^3}{6a}\right]_a^{3a} = \frac{13a^2}{3}$ | M1 A1 A1 |
$\text{Var}(X) = \frac{4a^2}{3}$ | M1 A1 A1 |
(b) $P(|X - \mu| < \sigma) = P(|X - 2a| < \frac{4a}{\sqrt{3}}) = \frac{2a}{\sqrt{3}} \times 2 \times \frac{1}{2a\sqrt{3}} = 0.577$ | M1 A1 A1 |
Normal: $P(|X - \mu| < \sigma) = P(|Z| < 1) = 2(0.3413) = 0.683$ | M1 A1 | **Total: 13**
---The random variable $X$ has a continuous uniform distribution on the interval $a \leq X \leq 3a$.
\begin{enumerate}[label=(\alph*)]
\item Without assuming any standard results, prove that $\mu$, the mean value of $X$, is equal to $2a$ and derive an expression for $\sigma^2$, the variance of $X$, in terms of $a$. [7 marks]
\item Find the probability that $|X - \mu| < \sigma$ and compare this with the same probability when $x$ is modelled by a Normal distribution with the same mean and variance. [6 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q5 [13]}}