type of a distribution function
Two distribution functions are said to of the same type if there exist such that . is called the scale parameter, and the location parameter or centering parameter. Let’s write to denote that and are of the same type.
Necessarily , for otherwise at least one of or would be violated.
If , then the graph of is shifted to the right from the graph of by units, if and to the left if .
If , then the graph of is stretched from the graph of by units if , and compressed if .
By the same token, we can classify all real random variables defined on a fixed probability space according to their distribution functions, so that if and are of the same type iff their corresponding distribution functions and are of type .
Given an equivalence class of distribution functions belonging to a certain type , such that a random variable of type exists with finite expectation and variance, then there is one distribution function of type corresponding to a random variable such that and . is called the standard distribution function for type . For example, the standard (cumulative) normal distribution is the standard distribution function for the type consisting of all normal distribution functions.
Within each type , we can further classify the distribution functions: if , then we say that and belong to the same location family under ; and if , then we say that and belong to the same scale family (under ).
|Title||type of a distribution function|
|Date of creation||2013-03-22 16:25:48|
|Last modified on||2013-03-22 16:25:48|
|Last modified by||CWoo (3771)|
|Defines||standard distribution function|