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I'm given a joint PDF defined as follows, $$ f(x,y) = \begin{cases} xy & 0 ≤ x ≤ y ≤ 1 \\ 0 &\mathrm{elsewhere} \end{cases} $$ I'm asked to find the marginal PDF and that's quite simple as I only have to integrate with respect to $x$ or $y$. But what I don't understand is the value of $x$ and $y$. Does the inequality mean that $x$ is somewhere between 0 and $y$ or $x$ is between 0 and 1? And also similarly for $y$.

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    1. This is not a PDF. 2. The inequalities mean that $0\le x\le y$ and $0\le y\le 1$ (or equivalently, that $x\le y\le1$ and $0\le x\le1$). – 2017-02-07
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    this means u take any (x,y) in $R^2$ then if f(x,y)>0(in sense of a neighburhood of(x,y)) x and y both lies in between 0 and 1 and x must be less than y – 2017-02-07
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    @Did I think it's ok to multiply by a constant to scale the enclosed volume to $1$, since it is only a density and not an absolute value, and there's sufficient info to work out that would be necessary. – 2017-02-08
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    @RobertFrost And I squarely disagree with this point of view. (The part "it is only a density and not an absolute value" seems to refer to nothing known on the subject, but please feel free to let it unexplained.) – 2017-02-08

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The first variable, $x$, ranges over $[0,y]$, with $y$, the second variable, ranging over $[0,1]$.

It would have been different with the inequality $$ 0 \le x,y \le 1. $$

Then, both variables would be ranging over $[0,1]$.