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Let $X,Y$ be two independent $Unif([0,1])$ random variables, denote $U=\min(X,Y)$, and $V=\max(X,Y)$. Find the joint density function of $U$ and $V$.

I'm trying to find the joint density function with $f_{U,V}(u,v)=f_{X,Y}(u,v)+f_{X,Y}(v,u)$ for $0<u<v<1$. But I don't know how to proceed from there. Any help will be greatly appreciated!

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1 Answer 1

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You have (removing the negligible events) for all non-negative measurable function $f$, $$ E(f(U,V))=E(f(X,Y)1_{X<Y})+E(f(Y,X)1_{Y<X})=2\int_{[0,1]^2} f(x,y)\,1_{x<y}\,dxdy $$ Hence the joint density function is $f_{U,V}(u,v)=21_{u<v}$.

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