How wide is the spread of random points?

Question

Scatter n independent points uniformly on [0, 1]. The range is the distance from the smallest point to the largest. What is the expected range, and what does it approach as n grows?

Accepted Answer

The range is R = X{(n)} - X{(1)}, the maximum minus the minimum. We can get its mean from the means of the two extreme order statistics. Using the equal-gaps picture (n points make n + 1 gaps, each of expected length \frac{1}{n+1}): the maximum sits at the top of n gaps and the minimum at the top of 1 gap: E[X{(n)}] = \frac{n}{n+1}, \qquad E[X{(1)}] = \frac{1}{n+1}. Therefore E[R] = \frac{n}{n+1} - \frac{1}{n+1} = \frac{n - 1}{n + 1}. Check small cases. For n = 2: E[R] = \frac{1}{3}, the expected distance between two random points, a classic result. For n = 5: E[R] = \frac{4}{6} = \frac{2}{3}…

How wide is the spread of random points?

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