Probability Space
A probability space is a measure space equipped with probability measure.
Contents
Description
A probability space is a triple (Ω, ℱ, P), composed of:
a sample space Ω
an event space ℱ, noting this is a σ algebra
a probability measure P : ℱ -> [0,1]
As for a probability measure, this is a map from ℱ to real numbers between 0 and 1. It must satisfy three conditions:
P(Ø) = 0
- additivity
see measure spaces for details
P(ℱ) = 1
Probability measures can theoretically take many forms, but conventionally they are probability distributions. Given a probability space (A, 𝒜, P) and a topological space (B, ℬ), a pushforward measure is defined as X : A -> B. X is both a random variable and its probability distribution. (Sometimes the topological space is instead described as a measurable space requiring that B is the Borel set of ℬ.)
In the case of a discrete random variable X, f is called a probability mass function and the final condition above is expressed as:
In the case of a continuous random variable X, f is called a probability density function and the final condition above is expressed as:
