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A and B are independent random variables, uniform distribution on $[0,1]$. What is $E(min(A,B))$ ?

$\displaystyle\int_{0}^{1}\int_{0}^{a}b\,db\,da + \displaystyle\int_{0}^{1} \int_{a}^{1}a\,db\,da$

$=\displaystyle\int_{0}^{1}\frac{a^2}{2}\,da+\int_{0}^{1}a-a^2\,da$

$=1/6+3/6-2/6=1/3$