Expected second smallest of six random numbers

Question

Draw 6 independent numbers uniformly from [0,1] and sort them. What is the expected value of the second smallest?

Accepted Answer

The k-th smallest of n independent uniforms averages \frac{k}{n+1}, since the points divide [0,1] into n+1 equal expected gaps. The second smallest is the case k = 2 with n = 6: E[	ext{second smallest}] = \frac{2}{6+1} = \frac{2}{7} \approx 0.286. By symmetry this mirrors the second largest, which would average \frac{6}{7}. Takeaway. Counting from the bottom uses the same rule \frac{k}{n+1}; the second smallest of n uniforms averages \frac{2}{n+1}.

Expected second smallest of six random numbers

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