What is the mean of max(U(0,1),U(0,1))? Judging by computer simulations, it must be at or around 2/3, but I have no idea how to compute the precise value.
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DYZ
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4I presume you a re assuming the 2 U[0,1] random variables are independent. Anyhow, see https://stats.stackexchange.com/questions/18433/how-do-you-calculate-the-probability-density-function-of-the-maximum-of-a-sample – Mark L. Stone May 20 '18 at 15:34
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Let $T$ be the quantity from OP, and let $x$ be another $U(0,1)$ variable.
The probability that $x > T$ is 1/3rd by symmetry. It is also $1 - E[T]$, by observation*. so we must have $E[T] = \frac{2}{3}$.
*$P(x \leq T) = \int T P(T) dT = E[T]$
shimao
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