Why isn't there a uniform probability distribution over the positive real numbers?

Solution 1:

If a uniform probability distribution did exist, then for any integer $n$ the probability a real number $x$ satisfies $n \leq x < n+1$ would have to be the same for all $n$. Call this probability $p$.

One of the rules that a probability distribution has to satisfy is that if $\{E_n\}$ are a countably infinite collection of disjoint events then $P(\cup_n E_n) = \sum_n P(E_n)$. Let $E_n$ be the event "the real number $x$ satisfies $n \leq x < n+1$". Since every real number is between some $n$ and $n + 1$, $P(\cup_n E_n) = 1$. On the other hand, $\sum_n P(E_n) = p + p + p + ....$. If $p > 0$, this gives infinity. If $p = 0$ it gives zero. In either case, you'll never add up to $1$. Hence you cannot have a uniform probability distribution over the reals.

Solution 2:

A uniform probability over the positive real numbers would not satisfy all three axioms of probability. Specifically, there is a conflict between the second axiom ($P(\Omega) = 1$) and the third axiom (countable additivity).

Check out the wikipedia entry http://en.wikipedia.org/wiki/Axioms_of_probability. I think it should be clear how a uniform probability over the positive reals would cause problems.

Solution 3:

For every probability density $f$, $\liminf_{x\to \pm \infty} f(x) = 0$ must hold.

It is clear that for a uniform distribution, the density has to be constant on the considered interval.

Combining both requirements, only $g(x) = 0$ remains as a choice for a uniform density on $(-\infty, \infty)$. But $\int_{-\infty}^\infty g(x)dx = 0 \neq 1$, that is $g$ is no proper probability density.

Solution 4:

"Uniform probability distribution" always refers to a pre-existent natural measure on a space with a transitive group of "translations", so that any two points resp. neighbourhoods of two points are comparable which each other. The simplest space of this sort would be the set $\mathbb Z$ of integers, and already there the problem you address becomes manifest. Of course it is easy to prove that there doesn't exist a uniform probability distribution on $\mathbb Z$; but you want an intuitive reasoning. While it is easy to come up with a random mechanism that selects any number between $-10^6$ and $10^6$ with equal probability, it is just unimaginable to fathom a mechanism that selects 5 with the same probability as any power of the prime with number $67212345$.