type of a distribution function

Two distribution functionsMathworldPlanetmath F,G:[0,1] are said to of the same type if there exist a,b such that G(x)=F(ax+b). a is called the scale parameter, and b the location parameter or centering parameter. Let’s write F=tG to denote that F and G are of the same type.


  • Necessarily a>0, for otherwise at least one of G(-)=0 or G()=1 would be violated.

  • If G(x)=F(x+b), then the graph of G is shifted to the right from the graph of F by b units, if b>0 and to the left if b<0.

  • If G(x)=F(ax), then the graph of G is stretched from the graph of F by a units if a>1, and compressed if a<1.

  • If X and Y are random variablesMathworldPlanetmath whose distribution functions are of the same type, say, F and G respectively, and related by G(x)=F(ax+b), then X and aY+b are identically distributed, since


    When X and aY+b are identically distributed, we write X=tY.

  • Again, suppose X and Y correspond to F and G, two distribution functions of the same type related by G(x)=F(ax+b). Then it is easy to see that E[X]< iff E[Y]<. In fact, if the expectation exists for one, then E[X]=aE[Y]+b. Furthermore, Var[X] is finite iff Var[Y] is. And in this case, Var[X]=a2Var[Y]. In general, convergence of moments is a “typical” property.

  • We can partitionMathworldPlanetmathPlanetmath the set of distribution functions into disjoint subsets of functions belonging to the same types, since the binary relationMathworldPlanetmath =t is an equivalence relationMathworldPlanetmath.

  • By the same token, we can classify all real random variables defined on a fixed probability spaceMathworldPlanetmath according to their distribution functions, so that if X and Y are of the same type τ iff their corresponding distribution functions F and G are of type τ.

  • Given an equivalence classMathworldPlanetmath of distribution functions belonging to a certain type τ, such that a random variable Y of type τ exists with finite expectation and varianceMathworldPlanetmath, then there is one distribution function F of type τ corresponding to a random variable X such that E[X]=0 and Var[X]=1. F is called the standard distribution function for type τ. For example, the standard (cumulative) normal distributionMathworldPlanetmath is the standard distribution function for the type consisting of all normal distribution functions.

  • Within each type τ, we can further classify the distribution functions: if G(x)=F(x+b), then we say that G and F belong to the same location family under τ; and if G(x)=F(ax), then we say that G and F belong to the same scale family (under τ).

Title type of a distribution function
Canonical name TypeOfADistributionFunction
Date of creation 2013-03-22 16:25:48
Last modified on 2013-03-22 16:25:48
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 13
Author CWoo (3771)
Entry type Definition
Classification msc 60E05
Classification msc 62E10
Synonym centering factor
Synonym scale parameter
Synonym location parameter
Defines type
Defines scale factor
Defines location factor
Defines standard distribution function
Defines location family
Defines scale family