Expected maximum of n exponentials

Question

Let X1, \dots, Xn be independent 	ext{Exp}(1) random variables (mean 1, CDF F(t) = 1 - e^{-t}). What is E[\max(X1, \dots, Xn)]?

Accepted Answer

Memoryless route (cleaner). The minimum of n independent 	ext{Exp}(1) variables is 	ext{Exp}(n), with expected value \frac{1}{n}: that is how long until the first one "fires." By the memoryless property, once it fires the remaining n-1 are still fresh 	ext{Exp}(1), so the wait for the next is 	ext{Exp}(n-1), mean \frac{1}{n-1}, and so on. The maximum is the time the last one fires, the sum of all these waits: E[\max] = \frac{1}{n} + \frac{1}{n-1} + \dots + \frac{1}{1} = Hn = \sum{j=1}^{n} \frac{1}{j}. CDF route (confirmation). P(\max \le t) = (1 - e^{-t})^n, so E[\max] = \int0^\infty \big…

Expected maximum of n exponentials

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