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, [ilmath]1[/ilmath], is spread evenly, or uniformly over the entire sample space, here denoted [ilmath]S[/ilmath], of a probability space here denoted [ilmath](S,\Omega,\mathbb{P})[/ilmath]

Discrete subset of [ilmath]\mathbb{N}_{\ge 0} [/ilmath]

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

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

Snippets

• for [ilmath]c\in\mathbb{R} [/ilmath] we define: [math]\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.[/math]