Distribution

2026-01-06

Probability

This page pairs well with Statistics.

Elements of Probability Theory

Definition (Random Experiment, Sample Space, Events)

A random experiment has uncertain outcomes. The sample space \(S\) is the set of all possible outcomes. An event \(E\) is a subset of \(S\). The certain event is \(S\); the impossible event is \(\varnothing\).

Definition (Probability Measure (Kolmogorov Axioms))

A probability space \((S,\mathcal{F},P)\) consists of a sample space \(S\), a \(\sigma\)-algebra \(\mathcal{F}\subseteq 2^S\), and a function \(P:\mathcal{F}\to[0,1]\) such that:

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