Prove there exists no uniform distribution on a countable and infinite set.

Can anyone help me with this problem, I can't figure out how to solve it...

Let $X$ be a random variable which can take an infinite and countable set of values. Prove that $X$ cannot be uniformly distributed among these values.

Thanks!

Solution 1:

Let $X$ be a random variable which assumes values in a countable infinite set $Q$. We can prove there is no uniform distribution on $Q$.

Assume there exists such a uniform distribution, that is, there exists $a \geq 0$ such that $P(X=q) = a$ for every $q \in Q$.

Observe that, since $Q$ is countable, by countable additivity of $P$,

$1 = P(X \in Q) = \sum_{q \in Q}{P(X = q)} = \sum_{q \in Q}{a}$

Observe that if $a=0$, $\sum_{q \in Q}{a}=0$. Similarly, if $a>0$, $\sum_{q \in Q}{a} = \infty$. Contradiction.

Solution 2:

If each point has probability $p\ne 0$, then there is some integer $n$ such that $np\gt 1$. So if $x_1,\dots,x_n$ are some of the distinct values taken on by $X$, the probability that $X=x_1$ or $X=x_2$ or $\dots$ or $X=x_n$ is $np$. Since $np\gt 1$, this is impossible.

On the other hand, if $p=0$, then by countable additivity the probability assigned to any subset of the values taken on by $X$ must be $0$, contradicting the fact that the sum of the probabilities over the range of $X$ must be $1$.