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Expected Maximum of Two Uniform Variables

Medium · Probability & Statistics · Quant Trader interview question · probability, expected-value, order-statistics, uniform-distribution

Two traders, Alice and Bob, each independently sample a random number from a uniform distribution between 0 and 1. Let Alice's number be $X$ and Bob's number be $Y$, where $X, Y \sim U(0, 1)$. What is the expected value of the larger of the two numbers, i.e., what is $E\max(X, Y) $?