Expected maximum of uniform random numbers

Question

You draw n independent numbers, each uniform on [0, 1] (think of n random priority scores), and keep the largest. What is the expected value of this maximum? Give the general formula and the n = 2 case.

Accepted Answer

Let M be the maximum of n i.i.d. 	ext{Uniform}[0,1] draws. CDF of the max. Since each draw satisfies P(Xi \le x) = x on [0,1] and they are independent, P(M \le x) = x^{n}. Expectation via the tail sum (continuous form). For a non-negative variable, E[M] = \int0^1 P(M x)\,dx = \int0^1 \big(1 - x^n\big)\,dx: E[M] = \left[x - \frac{x^{n+1}}{n+1}ight]0^1 = 1 - \frac{1}{n+1} = \frac{n}{n+1}. Case n = 2: E[M] = \frac{2}{3} \approx 0.667. By the same reasoning the minimum has P(\min x) = (1 - x)^n and E[\min] = \frac{1}{n+1}, so the two split the interval symmetrically: E[\min] + E[\max] = 1, and…

Expected maximum of uniform random numbers

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