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If $X$ and $Y$ are independent random variables and $P(X=c)P(Y=c) = 0$ for every $c$, what does it mean?

Does it mean X and Y are completely two different distributions? Also I interpret it as either both $P()=0$ or one of the $P()$ is equal to 0. For these two cases, what do they imply?

Ubanstan
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1 Answers1

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If they are continuous variables this is always the case.($\forall c\in \Bbb R:\: \Bbb P(X=c)=0$ if $X$ is continuous). Otherwise they have to have different distribution, because if they have the same then $\Bbb P(X=c)\Bbb P(Y=c)=\Bbb P(X=c)^2\neq 0$ for some $c$.

Dani
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