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Given a random event x from a discrete population X where the population $n=\infty$, and the population is uniformly distributed, what is P(x)?

My intuition is that it is infinitesimal, because as $lim_{n\to\infty}$, $ lim_{p\to0}$

Irongrave
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  • Related (and there are many other similar postings on this site): https://math.stackexchange.com/questions/1880140/is-getting-a-random-integer-even-possible – Ethan Bolker Jun 20 '17 at 00:14
  • Thanks, but that doesn't quite answer my question, and using search on the site didn't lead me anywhere useful for this particular inquiry. – Irongrave Jun 20 '17 at 00:23
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    In this context "infinitesimal" is the right intuition, but it's not precise. There is no uniform probability distribution on an infinite set. – Ethan Bolker Jun 20 '17 at 00:25
  • Sorry but I'm just not seeing the correlation -- why is there no uniform probability on an infinite set? – Irongrave Jun 20 '17 at 00:28

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There is no uniform discrete probability distribution on an infinite set $X$, for suppose $f:X\to\mathbb R$ is the probability mass function for such a distribution. Then $f$ should satisfy $$\sum_{x\in X}f(x)=1$$ and $$(\forall x,y\in X)(f(x)=f(y))$$ Pick $x_0\in X$. Then by the second requirement, for all $x\in X$ we have $f(x)=f(x_0)$. Then by the first requirement, $$\sum_{x\in X} f(x_0)=1$$ But this is impossible to satisfy. For if $f(x_0)=0$, then the sum comes out to zero; but if $f(x_0)\neq 0$, then the sum doesn't converge to a finite value.

florence
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