Let us say we have two continuous random variables, $X$ and $Y$ such that their pdfs $f(x)= f(-x)$ and $g(y)= g(-y)$ for all $x$ and $y$. In other words, $X$ and $Y$ have symmetric distributions around 0. Is it possible that joint distribution of $X$ and $Y$ will be such that $f_{joint}(x, y)$ is not always equal to $f_{joint}(-x, -y)$ for all $x$ and $y$? Please provide proof or simple counterexamples, if possible.


marked as duplicate by Dilip Sarwate, whuber Aug 15 '18 at 21:27

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    $\begingroup$ For all continuous random variables, $P(X=x)$ equals $0$ as does $P(X=-x)$. So your definition of symmetry has no meaning. $\endgroup$ – Dilip Sarwate Aug 15 '18 at 21:26
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    $\begingroup$ The duplicate answers your question. Note that it's easy to find discrete counterexamples, such as $(X,Y)$ with $\Pr(-1,-1)=\Pr(0,1)=1/3,\ \Pr(1,-1)=\Pr(1,1)=1/6.$ Just add a spherically symmetric continuous variable to $(X,Y)$ (such as a standard bivariate Normal) to produce a continuous counterexample. $\endgroup$ – whuber Aug 15 '18 at 21:32