Expected gap between the top two of five

Question

Draw 5 independent numbers uniformly from [0,1] and sort them. What is the expected gap between the largest and the second largest?

Accepted Answer

The k-th smallest of n uniforms averages \frac{k}{n+1}. With n = 5, the largest (5th smallest) averages \frac{5}{6} and the second largest (4th smallest) averages \frac{4}{6}. The expected gap between them is the difference: E[	ext{gap}] = \frac{5}{6} - \frac{4}{6} = \frac{1}{6} \approx 0.167. This matches the fact that all n+1 = 6 gaps between consecutive order statistics (and the two ends) have the same expected length \frac{1}{6}. Takeaway. Consecutive order statistics of n uniforms are spaced by the same average gap \frac{1}{n+1}, so the top two differ by \frac{1}{n+1} on average.

Expected gap between the top two of five

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