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I have a slight confusion with the notation, sometimes I find the joint density function like this: $$f_{XY}(x,y)$$ and on other occasions like this: $$f(x,y)$$ Is the same or what is the difference? Thanks!

JanetN
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  • So it would seem, consistently with the habit of indicating with $f_S$ the pdf of the random variable $S$. However, one might contend that $f_{X,Y}$ is more accurate, because $f_{XY}$ should be used for the pdf of the random variable $XY$, rather than for the pdf of the random vector $(X,Y)$. Notice that this ambiguity can be solved by observing that the pdf of $XY$ is a map $\Bbb R\to\Bbb R$, while the pdf of $(X,Y)$ is a map $\Bbb R^2\to\Bbb R$. –  Nov 05 '20 at 03:26

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In the case of $f_{XY}(x,y)$, you are specifying that the random variables you are taking the joint density of are $X$ and $Y$. If you just write $f(x,y)$, you had better be stating somewhere which random variables this joint density belongs to.

Robert Israel
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