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Let there be a real random number generator that absolutely randomly generates any number between 0 and infinity, that is, the entire series of natural numbers. For the overall probability to be normalized to 1, you need to divide one by infinity, and this is either zero or undefined. So what is the probability that the generator will generate any particular number?

Arman Armenpress
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  • @angryavian Why can't I? Is it impossible to create such a generator, or can it not generate very large numbers? What is the physical reason for the limitation?Also, the author of the second answer says it's possible:https://math.stackexchange.com/questions/41107/zero-probability-and-impossibility – Arman Armenpress Feb 12 '22 at 17:33
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    The Poisson distribution and geometric distribution are examples of probability distributions on the natural numbers, but they are not uniform. The argument in your question (also paraphrased here) shows why you cannot define a uniform distribution on the natural numbers without violating the axioms of probability. – angryavian Feb 12 '22 at 17:45
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    The answer by Fernando Martin that you mention is not relevant here, since he is discussing continuous distributions and you are talking about discrete distributions. – angryavian Feb 12 '22 at 17:46
  • @angryavian What's the difference? For example, the universe may well be infinite, with an infinite number of stars. Let's say all the stars are the same. Then the question "what is the probability that a randomly chosen star will be the Sun" is meaningless? It's a discrete distribution. – Arman Armenpress Feb 12 '22 at 17:53
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    A uniform distribution on natural numbers does not exist. This because no $p\geq0$ exists with $\sum_{n=1}^{\infty}p_n=1$ and $p_n=p$ for every $n$ at the same time. Also on set $[0,\infty)$ there is no uniform distribution. This because no constant $c\geq0$ exists with $\int_0^{\infty}cdx=1$. – drhab Feb 12 '22 at 19:05
  • @drhab And the example with stars written above can be transferred to a continuous distribution? – Arman Armenpress Feb 12 '22 at 19:10
  • If the set of stars is countable then a continuous distribution on it is not possible. If it is uncountable then a continuous distribution might be possible but not a uniform continuous distribution. – drhab Feb 12 '22 at 19:52
  • @drhab It is believed that the number of stars is infinite. You can simply make the universe finite, and consider each star as a point of zero size on its surface. There will be a continuous distribution, the probability for each point will be zero. – Arman Armenpress Feb 12 '22 at 20:01
  • @ArmanArmenpress If we believe whether the number of stars is infinite or if the are even points in a continuum is not a math question. – Kurt G. Feb 13 '22 at 09:35

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