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I have this probability density of $f(x,y)$ if $0\le x \lt y\le 1$ and I can't seem to get the right answer. It says in the back of my book that the bound's on the integral for say $f_X(x)$ is from $0$ to $x$. Shouldn't the integral to find the marginal density of $X$ be $f_X(x)=\int_x^1 f(x,y)dy$ Then it says the marginal for $Y$ is $f_Y(y)=\int_y^1f(x,y)dx$ Shouldnt this one be $f_Y(y)=\int_0^yf(x,y)dx$?? Im slightly confused now. Is it because $x$ is strictly less than $y$?

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    @DilipSarwate Thanks that's what I thought. I knew that didn't look right. So glad I have a mathematical community to correct me!2012-12-21

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If $f(x,y)$ is the joint density of the random variables $X,Y$, then the marginal density of $X$ is $ f_X(x)=\int_{\mathbb{R}}{f(x,y)dy} $ Now, you know that $f(x,y)$ equals zero if $y, or $x$ or $y$ falls outside the interval $[0,1]$ (or so I interpret your question) and thus the formula simplifies to $ f_X(x)=\int_x^1{f(x,y)dy}. $

Similarly one finds $ f_Y(y)=\int_0^y{f(x,y)dx}. $

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    Thank you that clears it up. My book has an error.2012-12-21