Topic: Probability and Statistics
Find the variance of the distribution shown in the figure.
- A. $\frac{\Delta^2}{2}$
- B. $\frac{\Delta^2}{12}$
- C. $\frac{\Delta^2}{8}$
- D. $\frac{\Delta^2}{4}$
Answer: B. $\frac{\Delta^2}{12}$
Explanation: The figure shows a continuous uniform distribution from $-\frac{\Delta}{2}$ to $\frac{\Delta}{2}$ with a constant probability density function $p(x) = \frac{1}{\Delta}$. The variance $\sigma^2$ of a uniform distribution over the interval $[a, b]$ is given by $\frac{(b-a)^2}{12}$. Here, the interval width is $\Delta$, so the variance is $\frac{\Delta^2}{12}$.