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Is there any difference between the two? I have not met any formal definition of the support of a random variable. I know that for the function $f$ the support is a closure of the set $\{y:\;y=f(x)\ne0\}$.

Bayes
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The support of the probability distribution of a random variable $X$ is the set of all points whose every open neighborhood $N$ has the property that $\Pr(X\in N)>0$.

It is more accurate to speak of the support of the distribution than that of the support of the random variable.

The complement of the support is the union of all open sets $G$ such that $\Pr(X\in G)=0$. Since the complement is a union of open sets, the complement is open. Therefore the support is closed.

Michael Hardy
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  • I see. It is often written "let $X$ be a random variable with support S", which confused me, since the random variable is also a function. Thank you for clarification! – Bayes Jun 10 '13 at 02:33
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    Note that the support of the distribution of $X$ is the same as the essential range of the measurable function $X$ itself. – Nate Eldredge Jun 10 '13 at 02:54
  • What about measure theory? I believe the definition of the support of a function $f : A \rightarrow B $ is : {$x \in A : f(x) \neq 0$}, that is the same as the range of the function, right? – Frank Aug 13 '19 at 13:15
  • This doesn't distinguish between the two. Here the support is defined but the range is not defined. Would welcome a definition of range to clarify. – socialscientist May 13 '23 at 00:13