Probability space

From CS2800 wiki

Sample space, event, outcome

Definition: Sample space
A sample space S is just a set. We refer to the elements of S as outcomes. We refer to the subsets of S as events.

Probability measure, Kolmogorov's axioms

A probability measure on a sample space S is a function [math]\href{/cs2800/wiki/index.php/Pr}{Pr} : \href{/cs2800/wiki/index.php/2}{2}^\href{/cs2800/wiki/index.php/S}{S} \href{/cs2800/wiki/index.php/%E2%86%92}{→} \href{/cs2800/wiki/index.php?title=%E2%84%9D&action=edit&redlink=1}{ℝ} [/math] satisfying the following three properties:
  1. For all events [math]E [/math], [math]\href{/cs2800/wiki/index.php/Pr}{Pr}(E) ≥ 0 [/math]
  2. [math]\href{/cs2800/wiki/index.php/Pr}{Pr}(S) = 1 [/math]
  3. For all disjoint events [math]E_1 [/math] and [math]E_2 [/math], [math]\href{/cs2800/wiki/index.php/Pr}{Pr}(E_1 \href{/cs2800/wiki/index.php/%E2%88%AA}{∪} E_2) = \href{/cs2800/wiki/index.php/Pr}{Pr}(E_1) + \href{/cs2800/wiki/index.php/Pr}{Pr}(E_2) [/math]
These three properties are referred to as the Kolmogorov axioms.

Probability space

Definition: Probability space
A probability space is just a pair containing a sample space S and a probability measure [math]\href{/cs2800/wiki/index.php/Pr}{Pr} [/math] on S.