Expected maximum of uniform random variables

Question

Let X and Y be independent U(0,1) random variables. What are E[\max(X,Y)] and E[\min(X,Y)]? Generalize to the maximum of n independent uniforms.

Accepted Answer

CDF method. For M = \max(X, Y) and t \in [0,1], P(M \le t) = P(X \le t)P(Y \le t) = t^2, so the density is fM(t) = 2t and E[M] = \int0^1 t \cdot 2t \, dt = \frac{2}{3}. Min via symmetry. Since \max + \min = X + Y, E[\min] = E[X] + E[Y] - E[\max] = \frac12 + \frac12 - \frac23 = \frac{1}{3}. General n. With Mn = \max(X1, \dots, Xn), P(Mn \le t) = t^n, so E[Mn] = \int0^1 t \cdot n t^{n-1} dt = \frac{n}{n+1}, and by the mirror symmetry t \mapsto 1 - t, E[\minn] = \frac{1}{n+1}. Intuition. n uniform points split [0,1] into n + 1 gaps that are exchangeable, so each gap has expected length \frac{1}{…

Expected maximum of uniform random variables

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