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Let $X, Y$ be random variables with density functions $f_X(x)$ and $f_Y(y)$.

If $X$ and $Y$ are independent, then the joint density function $f(x,y)=f_X(x)f_Y(y)$.

If they are not independent, how can we find the joint density function(if it exists)?

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We can't. Say $X$ and $Y$ are both uniform on $[0,1]$. Then if they are independent, $(X,Y)$ is uniform on the unit square. If, on the other hand, $X=Y$, then $(X,Y)$ is uniform on the line segment from $(0,0)$ to $(1,1)$. One can certainly find more exotic joint distributions too with the same component-wise distributions.

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