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Let $X,Y$ be independent and uniformly distributed in the interval $(0,1)$. What is the joint probability density function? The answer I get is: $$f_{X,Y}(x,y) = 1,\quad 0<x<1,\quad 0<y<1$$

Is this correct? My intuition here is completely off for some reason and I'm really confused.

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Yes, you are correct.

$$f_{X,Y}(x,y)=f_X(x)f_Y(y)$$

since they are independent.

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