nLab probability distribution




A probability distribution is a measure used in probability theory whose integral over some subspace of a measurable space is regarded as assigning a probability for some event to take values in this subset.

Often a probability density.


On measurable spaces

A probability distribution is a measure ρ\rho on a measurable space XX such that

  • it is positive: UX: Udρ0\forall U \subset X : \int_U d\rho \geq 0;

  • it is normalized: Xdρ=1\int_X d\rho = 1.

On σ\sigma-frames

In measure theory, a probability measure or probability distribution on a σ \sigma -frame or more generally a σ \sigma -complete distributive lattice (L,,,,,,)(L, \leq, \bot, \vee, \top, \wedge, \Vee) is a probability valuation μ:L[0,1]\mu:L \to [0, 1] such that the elements are mutually disjoint and the probability valuation is denumerably/countably additive

sL .m.n.(mn)(s(m)s(n)=)\forall s\in L^\mathbb{N}. \forall m \in \mathbb{N}. \forall n \in \mathbb{N}. (m \neq n) \wedge (s(m) \wedge s(n) = \bot)
sL .μ( n:s(n))= n:μ(s(n))\forall s\in L^\mathbb{N}. \mu(\Vee_{n:\mathbb{N}} s(n)) = \sum_{n:\mathbb{N}} \mu(s(n))


The collection of all probability distributions on a measurable space carries various metric structures that are studied in information geometry:



category: probability

Last revised on February 7, 2024 at 17:04:32. See the history of this page for a list of all contributions to it.