Probability distribution (countable sample space)

Definition

A prob­a­bil­ity dis­tri­bu­tion on a countable Sam­ple space \(\Omega\) is a Func­tion \(\mathbb{P}: \Omega \to [0,1]\) such that \(\sum_{\omega \in \Omega} \mathbb{P}(\omega) = 1\).

Intuition

We ex­press a be­lief that ”\(x\in \Omega\) hap­pens with prob­a­bil­ity \(r\)” by set­ting \(\mathbb{P}(x) = r\). So a prob­a­bil­ity dis­tri­bu­tion di­vides up our an­ti­ci­pa­tion of what will hap­pen, out of the set \(\Omega\) of things that might pos­si­bly hap­pen.

ex­am­ples, futher points

Parents:

  • Probability theory

    The logic of sci­ence; co­her­ence re­la­tions on quan­ti­ta­tive de­grees of be­lief.