type of a distribution function
Two distribution functions F,G:ℝ→[0,1] are said to of the same type if there exist a,b∈ℝ such that G(x)=F(ax+b). a is called the scale parameter, and b the location parameter or centering parameter. Let’s write Ft=G to denote that F and G are of the same type.
Remarks.
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Necessarily a>0, for otherwise at least one of G(-∞)=0 or G(∞)=1 would be violated.
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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.
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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.
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If X and Y are random variables
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
P(X≤z)=F(z)=G(z-ba)=P(Y≤z-ba)=P(aY+b≤z). When X and aY+b are identically distributed, we write Xt=Y.
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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.
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We can partition
the set of distribution functions into disjoint subsets of functions belonging to the same types, since the binary relation
t= is an equivalence relation
.
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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 X and Y are of the same type τ iff their corresponding distribution functions F and G are of type τ.
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Given an equivalence class
of distribution functions belonging to a certain type τ, such that a random variable Y of type τ exists with finite expectation and variance
, 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 distribution
is the standard distribution function for the type consisting of all normal distribution functions.
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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 |