Highest of Three

Question

Three numbers are drawn independently and uniformly from [0,1]. What is the expected value of the largest of the three?

Accepted Answer

The maximum is below x exactly when all three draws are: F_max(x) = P(max le x) = x^3, qquad x in [0,1]. Then, using the tail formula for a variable supported on [0,1]: mathbbE[max] = int_0^1 left(1 - x^3right) dx = 1 - frac14 = frac34. Equivalently, differentiate: density 3x^2, so mathbbE = int_0^1 3x^3,dx = frac34. Approach 2, the gaps picture Three uniform points split [0,1] into four gaps which are exchangeable, so each has expected length frac14. The maximum sits at the end of the first three gaps: mathbbE[max] = frac34. This symmetry argument instantly gives the general k-th smallest of n draws: mathbbE[X_(k)] = frackn+1, with no integration at all. Why it matters: order statistics of uniforms are the skeleton for auction intuition (highest of n bids), best-of-n quotes, and percentile estimates. The frackn+1 rule is one of the highest-value formulas to have memorized, and interviewers often accept the gaps argument as the "real" answer.

Highest of Three

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