Given $f(x,y)$ a positive smooth function on $[0,1] \times [0,1]$ that satisfies the condition$\int f(x,y) dx = \int f(y,x) dx$ for all $y \in [0,1]$.
Is it true that $f(x,y) = f(y,x)$?
If not, could you please provide a counter example?
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Question on the symmetry of the joint density of two identically distributed random variables. [closed]
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