probability distribution function


1 Definition

Let (Ω,𝔅,μ) be a measure spaceMathworldPlanetmath. A probability distribution function on Ω is a function f:Ω⟶ℝ such that:

  1. 1.

    f is μ-measurable

  2. 2.

    f is nonnegative μ-almost everywhere.

  3. 3.

    f satisfies the equation

    ∫Ωf⁢(x)⁢𝑑μ=1

The main feature of a probability distribution function is that it induces a probability measure P on the measure space (Ω,𝔅), given by

P⁢(A):=∫Af⁢(x)⁢𝑑μ=∫Ω1A⁢f⁢(x)⁢𝑑μ,

for all A∈𝔅. The measure P is called the associated probability measure of f. Note that P and μ are different measures, though they both share the same underlying measurable spaceMathworldPlanetmathPlanetmath (Ω,𝔅).

2 Examples

2.1 Discrete case

Let I be a countable set, and impose the counting measure on I (μ⁢(A):=|A|, the cardinality of A, for any subset A⊂I). A probability distribution function on I is then a nonnegative function f:I⟶ℝ satisfying the equation

∑i∈If⁢(i)=1.

One example is the Poisson distributionMathworldPlanetmath Pr on ℕ (for any real number r), which is given by

Pr⁢(i):=e-r⁢rii!

for any i∈ℕ.

Given any probability space (Ω,𝔅,μ) and any random variableMathworldPlanetmath X:Ω⟶I, we can form a distribution functionMathworldPlanetmath on I by taking f(i):=μ({X=i}). The resulting function is called the distribution of X on I.

2.2 Continuous case

Suppose (Ω,𝔅,μ) equals (ℝ,𝔅λ,λ), the real numbers equipped with Lebesgue measureMathworldPlanetmath. Then a probability distribution function f:ℝ⟶ℝ is simply a measurable, nonnegative almost everywhere function such that

∫-∞∞f⁢(x)⁢𝑑x=1.

The associated measure has Radon–Nikodym derivativePlanetmathPlanetmath (http://planetmath.org/RadonNikodymTheorem) with respect to λ equal to f:

d⁢Pd⁢λ=f.

One defines the cumulative distribution functionMathworldPlanetmath F of f by the formulaMathworldPlanetmathPlanetmath

F(x):=P({X≤x})=∫-∞xf(t)dt,

for all x∈ℝ. A well known example of a probability distribution function on ℝ is the Gaussian distribution, or normal distributionMathworldPlanetmath

f⁢(x):=1σ⁢2⁢π⁢e-(x-m)2/2⁢σ2.
Title probability distribution function
Canonical name ProbabilityDistributionFunction
Date of creation 2013-03-22 12:37:25
Last modified on 2013-03-22 12:37:25
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 11
Author Mathprof (13753)
Entry type Definition
Classification msc 60E99
Synonym probability density function
Synonym distribution
Related topic Measure
Related topic Stochastic
Related topic DiscreteDensityFunction
Related topic DistributionFunction
Related topic DensityFunction
Related topic AreaUnderGaussianCurve
Defines cumulative distribution function