nLab probability distribution

Idea

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.

Definition

On measurable spaces

A probability distribution is a measure $\rho$ on a measurable space $X$ such that

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

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

On $\sigma$-frames

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

$$\forall s\in L^\mathbb{N}. \forall m \in \mathbb{N}. \forall n \in \mathbb{N}. (m \neq n) \wedge (s(m) \wedge s(n) = \bot)$$

$$\forall s\in L^\mathbb{N}. \mu(\Vee_{n:\mathbb{N}} s(n)) = \sum_{n:\mathbb{N}} \mu(s(n))$$

Properties

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

• information metric

• Wasserstein metric

Examples

• Gaussian probability distribution

Related concept

• probability space

• expectation value

• moment

• entropy

• stochastic map

• positive-operator valued probability measure

• density of a subset

• unit interval

• sigma-frame

• sigma-complete lattice

• distributive lattice

• probability valuation

• sigma-continuous probability valuation

References

• Alex Simpson, Measure, randomness and sublocales.

category: probability