Show that there is no discrete uniform distribution on N.

Question

This is a homework question I got. I'm not entirely sure what it is asking. Can someone please clarify/get me on the right track?

Answer 1

In general you can indeed have a probability distribution on $\mathbb{N}$. For instance, if $X$ is such a random variable we can define a probability mass function for each natural number by assigning a number for each probability, that is $P(X=n)=a_n$ with $0\leq a_n\leq 1$ and require that $$\sum_{n\geq 0} P(X=n)= \sum_{n\geq 1} a_n = 1.$$

Observe that the sum is infinite (over all Natural numbers) and convergent, hence by Cauchy's criterion we must have $\lim_{n\to 0} a_n = 0$.

Now if we want $X$ to be uniformly distributed, which means $P(X=n)= P(X=m)$ for all $n,m\in \mathbb{N}$ then we need $a_n=a_m$ for all $n,m\in \mathbb{N}$. Call $k:=a_n$ such value. But then $$\sum_{n\geq 1} a_n = \sum_{n\geq 1} k = \infty \neq 1$$ so such distribution function can not be a probability function. Hence, it is impossible to assign equal probabilities to all natural numbers and expect that the sum of all these probabilities equals 1.