prove that X and Y are independent if and only if f_{x|y}(x|y)= f_{x} (x) for all x and y.
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$\displaystyle f_{X|Y}(x|y)=\frac{f_{X,Y}(x,y)}{f_Y(y)}$ $\displaystyle X$ and $\displaystyle Y$ independent $\displaystyle \implies f_{X,Y}(x, y)=f_X(x)f_Y(y)$
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