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The first comment under the question says it has 'non-zero probability'. I don't see how that is the case, as 1 divided by countable infinity should still be 0, which is same in value as 1 divided by uncountable infinity. Also I think the conclusion should be the same regardless of whether its domain is a finite interval or not.

Sam
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    Beware of working with and reasoning about infinities as if they were finite numbers. A concrete example of such a random variable might help you out: see our thread at https://stats.stackexchange.com/questions/103969. – whuber Oct 04 '23 at 20:55
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    You cannot have a uniform distribution over a countably infinite set, so there is no need to think about "1 divided by countable infinity" – Henry Oct 04 '23 at 21:18
  • @Henry so the probability mass is zero everywhere right? – Sam Oct 05 '23 at 01:39
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    @Sam, a uniform distribution over a countably infinite set cannot exist, so it is difficult to talk about a probability mass or probability density of something which is not. If a measure of each point in the set were a constant $m$ then you would get a total of $\sum m=\infty$ if $m$ is positive or $\sum m=0$ if $m$ is zero (or perhaps $\sum m=-\infty$ if you allowed $m$ negative), so this would not be a probability measure where you need $\sum m=1$. No division required. – Henry Oct 05 '23 at 08:26

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