= Probability Space = A '''probability space''' is a [[Analysis/MeasureSpace|measure space]] equipped with probability measure. <> ---- == Description == A probability space is a triple ''(Ω, ''ℱ'', P)'', composed of: * a sample space ''Ω'' * an event space ℱ, noting this is a [[Analysis/SigmaAlgebra|σ algebra]] * a '''probability measure''' ''P :'' ℱ ''-> [0,1]'' satisfying three conditions called the [[Analysis/KolmogorovAxioms|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 [[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''.) In the case of a discrete random variable ''X'', ''f'' is called a '''probability mass function''' and the final condition above is expressed as: {{attachment:disc.svg}} In the case of a continuous random variable ''X'', ''f'' is called a '''probability density function''' and the final condition above is expressed as: {{attachment:cont.svg}} ---- CategoryRicottone