TITLE: Expectation of the min of two independent random variables?
QUESTION [30 upvotes]: How do you compute the minimum of two independent random variables in the general case ?
In the particular case there would be two uniform variables with a difference support, how should one proceed ?
EDIT: specified that they were independent and that the uniform variables do not have obligatory the same support range.

REPLY [5 votes]: Let: $U = \min(X,Y)$, where $\min(X,Y)\leq z$.
$Pr(\min(X,Y) > z) = Pr((X>z) \cap (Y>z))$.
$Pr(U>z) = Pr(X>z)*Pr(Y>z)$.
$Pr(U\geq z) = (1 - Fx(z))*(1 - Fy(z))$.
$Fu(z) = 1 - (1 - Fx(z))*(1 - Fy(z))$.
Thus,
$F_\min(x,y) = Fx(z) + Fy(z) - Fx(z)*Fy(z)$.