Does it exists probability space $\Omega = (N, \mathcal{P}(A), P)$ that $P({n}) = const$ for $n \in N$?

Asked Oct 19 '21 at 20:37

Active Oct 20 '21 at 18:34

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I'm started my probability course two weeks ago and misunderstand how to approach to the following problem

Does it exists probability space $\Omega = (\mathbb N, \mathcal{P}(\mathbb N), P)$ that $P(\{n\}) = const$ for $n \in \mathbb N$ ?

M b I can not understand the essence of this task.

But how should I start?

I have two ideas.

The first one is: if $P(${$n$}$) = const$ $\Rightarrow$ $\sum P(${$n$}$) = 1$ $\Rightarrow$ $const \rightarrow 0$ and we have contradiction.

And the second one: Assume that if $n$ is even $P(${$N$}$) = 0.5$, if uneven $P(${$N$}$) = 0.5$

Thank u

edited Oct 20 '21 at 18:34

Mike Earnest

asked Oct 19 '21 at 20:37

Ivan Dumanov

Is $N$ shorthand for ${1, \ldots, N}$? If so, the uniform distribution would satisfy your requirement. – Oct 19 '21 at 21:04
Your first idea works. I don't understand your second idea. – user6247850 Oct 19 '21 at 21:08
If $N$ is shorthand for all the natural numbers, then no - there is no uniform distribution on the natural numbers – Henry Oct 19 '21 at 21:11
$\mathcal{N}$ is the infinite set of Natural numbers – Ivan Dumanov Oct 20 '21 at 07:19

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