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For the joint density function $$P\big((X,Y) \in A\big) = \int_A f_{(X,Y)} (x,y) \, dx\,dy$$ how would you show that if $(X,Y)$ is a random vector in $\mathbb{R}^2$ with density $f_{(X,Y)}$ and $f_{(X,Y)}(x,y) = f(x)g(y)$ for a pair of non-negative functions $f$ and $g$ then $X$ has density $$\frac{f}{\int_\mathbb{R}f(t)\,dt}$$ and $Y$ has density $$\frac{f}{\int_\mathbb{R}g(t)\,dt}\, \, ?$$

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    If I understand correctly, X and Y would be independent, f and g would be their marginal distributions and would thus be their density functions as well. Integrated over R, f and g would yield an answer of unity and the statement would hold but this doesn't seem right for some reason. Maybe someone will point out why.2012-10-17
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    The only thing that, it is not said that $f$ and $g$ are already density functions, so they may differ in a constant $c=\int_{\Bbb R}f$..2012-10-17

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