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| * a '''probability measure''' ''P :'' ℱ ''-> [0,1]'' | * a '''probability measure''' ''P :'' ℱ ''-> [0,1]'' satisfying three conditions called the [[Analysis/KolmogorovAxioms|Kolmogorov axioms]] |
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| As for a probability measure, this is a [[Analysis/Functions|map]] from ℱ to real numbers between 0 and 1. It must satisfy three conditions: * ''P(Ø) = 0'' * additivity * see [[Analysis/MeasureSpace|measure spaces]] for details * ''P(''ℱ'') = 1'' |
A closely related concept is a '''random variable''', which itself is just a map between a sample space and some measurable space ''(Ω, ''𝒜'')'': ''X : Ω ->'' 𝒜. When the sample space ''Ω'' is common to both the random variable and a probability space, it is possible to lift the probability measure ''P'' from the latter into the former. This '''pushforward measure''' is a probability distribution. The probability that the random variable ''X'' takes on the value ''x ∈ '' 𝒜 is given by ''P(X = x) = P({ω ∈ Ω | X(ω) = x})''. The probability that the random variable takes on any value in the subset ''S ⊆'' 𝒜 is given by ''P(X ∈ S) = P({ω ∈ Ω | X(ω) ∈ S})''. |
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| Probability measures can theoretically take many forms, but conventionally they are '''probability distributions'''. Given a probability space ''(A, ''𝒜'', P)'' and a [[Analysis/TopologicalSpace|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 [[Analysis/SigmaAlgebra|Borel set]] of ℬ.) | (It may be more appropriate to say that the random variable maps to a [[Analysis/TopologicalSpace|topological space]]. Practically speaking, probability is applied in two ways: the sample space is countable and discrete, or the sample space is uncountable but the random variable maps to real numbers. In either case, the [[Analysis/SigmaAlgebra|Borel set]] of ''Ω'' is the smallest possible [[Analysis/SigmaAlgebra|σ algebra]] that can be defined on ''Ω''.) |
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] satisfying three conditions called the Kolmogorov axioms
A closely related concept is a random variable, which itself is just a map between a sample space and some measurable space (Ω, 𝒜): X : Ω -> 𝒜. When the sample space Ω is common to both the random variable and a probability space, it is possible to lift the probability measure P from the latter into the former. This pushforward measure is a probability distribution. The probability that the random variable X takes on the value x ∈ 𝒜 is given by P(X = x) = P({ω ∈ Ω | X(ω) = x}). The probability that the random variable takes on any value in the subset S ⊆ 𝒜 is given by P(X ∈ S) = P({ω ∈ Ω | X(ω) ∈ S}).
(It may be more appropriate to say that the random variable maps to a topological space. Practically speaking, probability is applied in two ways: the sample space is countable and discrete, or the sample space is uncountable but the random variable maps to real numbers. In either case, the Borel set of Ω is the smallest possible σ algebra that can be defined on Ω.)
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:
