# Ways of Constructing a Non-Uniform Probability Space

I want to construct a non-uniform probability space $(\Omega, P)$ and a random variable $X$ on this space such that $X$ is uniformly distributed over a set of 4 values. And I want to make $\Omega$ as small as possible.

Is there a way to make such a space(s) using more than just trial and error. Is there a systematic or intuitive way in which one could go about making this space?

• If $X$ takes $4$ distinct values, then there must be at least $4$ elements in $\Omega$. – Math1000 Oct 30 '17 at 5:14

## 1 Answer

For each of the four values, the set of points where $X$ has this value must be nonempty. So there must be at least four points. If there are only four points in the probability space, the probability of each of these points must correspond to a unique value of $X$, so the distribution would have to be uniform. So we need more points. If you have a probability space with five points, there must be exactly one value that is achieved on two points. Still, you can induce a uniform distribution over the four values. But not by a uniform distribution over the five points.