Do the random points fit in a short window?

Question

n points are dropped independently and uniformly on the interval [0, 1]. What is the probability that all n fit inside some window of length w (that is, the range max minus min is at most w)? Evaluate n = 3, w = 	frac12.

Accepted Answer

Condition on which point is the minimum. Any of the n points can be the leftmost (a factor n). Given the minimum sits at position x, the other n - 1 points must lie in [x, x + w], but also cannot exceed the segment's right end at 1. So they must all fall in an interval of length \min(w,\, 1 - x), which happens with probability \big(\min(w, 1-x)\big)^{n-1}. Integrating over x: P(	ext{range} \le w) = n \int0^1 \big(\min(w,\, 1-x)\big)^{n-1}\,dx. Split at x = 1 - w. For x \le 1 - w the window fits and the integrand is w^{n-1}; for x 1 - w the segment's edge clips it to 1 - x: P = \underbrace{n\…

Do the random points fit in a short window?

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