View source for 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, {{M|1}}, is spread evenly, or ''uniformly'' over the entire {{link|sample space|probability}}, here denoted {{M|S}}, of a [[probability space]] here denoted {{M|(S,\Omega,\mathbb{P})}}
===Discrete subset of {{M|\mathbb{N}_{\ge 0} }}===
We will cover the common cases, and their notation, first:
* for {{M|a,b\in\mathbb{N}_{\ge 0} }} we have: {{M|X\sim\text{Uni}(a,b)}} to mean:
** We form a [[probability space]], {{M|(S,\Omega,\mathbb{P})}}
*** {{M|S:\eq\{a,a+1,\ldots,b-1,b\}\subseteq\mathbb{N}_{\ge 0} }}
*** {{M|\Omega:\eq}}[[Sigma-algebra generated by|{{M|\sigma(S)}}]] which in this case means: {{M|\Omega\eq}}[[Power set|{{M|\mathcal{P}(S)}}]]

==Snippets==
*for {{M|c\in\mathbb{R} }} we define: {{MM|\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.}}