Let $f(x,y)=2x$ for $0 < x < y < 1$ and zero otherwise. I would like to find the marginal distribution $f_{X}(x)$, which is equal to $$\int f(x,y)dy.$$ However, should integrate over $[0, y]$ or $[0, 1]$? Why?
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$\begingroup$ For fixed $x$, what are the possible values for $y$? $\endgroup$– dsaxtonCommented Dec 8, 2015 at 20:35
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$\begingroup$ So if $x$ is fixed, it could be anything between $0$ and $y$. So when taking the integral, I should integrate over $[x, 1]$, since $x<y$ and $y<1$? $\endgroup$– user97572Commented Dec 8, 2015 at 20:40
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$\begingroup$ $y \in (x, 1)$ for fixed $x$. $\endgroup$– dsaxtonCommented Dec 8, 2015 at 21:15
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1 Answer
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You need to integrate over the support of $y$ which in this case is $(x,1)$ by the definition of the bounds on $x$ and $y$ given in the question.
To be explicit, in order to obtain the marginal distribution of $X$ you need to solve the following:
$$f_X(x)=\int f_{X,Y}(x,y)dy=\int_x^1 2x dy$$