Uniform probability distribution

Definition

There are a few distinct cases we may define the uniform distribution on, however in any case the concept is clear:

The total probability, $1$, is spread evenly, or uniformly over the entire sample space, here denoted $S$, of a probability space here denoted $(S,\Omega,\mathbb{P})$

Discrete subset of $\mathbb{N}_{\ge 0}$

We will cover the common cases, and their notation, first:

• for $a,b\in\mathbb{N}_{\ge 0}$ we have: $X\sim\text{Uni}(a,b)$ to mean:
  • We form a probability space, $(S,\Omega,\mathbb{P})$
    • $S:\eq\{a,a+1,\ldots,b-1,b\}\subseteq\mathbb{N}_{\ge 0}$
    • $\Omega:\eq$$\sigma(S)$ which in this case means: $\Omega\eq$$\mathcal{P}(S)$

Snippets

• for $c\in\mathbb{R}$ we define: $$\mathbb{P}[X\eq c]:\eq\left\{\begin{array}{lr}\frac{1}{(b-a)+1} & \text{for }c\in\{a,\ldots,b\}\subseteq\mathbb{N}_{\ge 0} \\0 & \text{otherwise}\end{array}\right.$$