distribution function

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Let F:. Then F is a distribution functionMathworldPlanetmath if

  1. 1.

    F is nondecreasing,

  2. 2.

    F is continuous from the right,

  3. 3.

    limx-F(x)=0, and limxF(x)=1.

As an example, suppose that Ω= and that is the σ-algebra of Borel subsets of . Let P be a probability measureMathworldPlanetmath on (Ω,). Define F by


This particular F is called the distribution function of P. It is easy to verify that 1,2, and 3 hold for this F.

In fact, every distribution function is the distribution function of some probability measure on the Borel subsets of . To see this, suppose that F is a distribution function. We can define P on a single half-open interval by


and extend P to unions of disjoint intervals by


and then further extend P to all the Borel subsets of . It is clear that the distribution function of P is F.

0.1 Random Variables

Suppose that (Ω,,P) is a probability space and X:Ω is a random variableMathworldPlanetmath. Then there is an induced probability measure PX on defined as follows:


for every Borel subset E of . PX is called the distributionPlanetmathPlanetmath of X. The distribution function of X is


The distribution function of X is also known as the law of X. Claim: FX = the distribution function of PX.

FX(x) = P(ω|X(ω)x)
= P(X-1((-,x])
= PX((-,x])
= F(x).

0.2 Density Functions

Suppose that f: is a nonnegative function such that


Then one can define F: by


Then it is clear that F satisfies the conditions 1,2,and 3 so F is a distribution function. The function f is called a density function for the distribution F.

If X is a discrete random variable with density function f and distribution function F then

Title distribution function
Canonical name DistributionFunction
Date of creation 2013-03-22 13:02:51
Last modified on 2013-03-22 13:02:51
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 16
Author Mathprof (13753)
Entry type Definition
Classification msc 60E05
Synonym cumulative distribution functionMathworldPlanetmath
Synonym distribution
Related topic DensityFunction
Related topic CumulativeDistributionFunction
Related topic RandomVariable
Related topic Distribution
Related topic GeometricDistribution2
Defines law of a random variable