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type of a distribution function
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(Definition)
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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.
Remarks.
- 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 .
- If
and are random variables whose distribution functions are of the same type, say, and respectively, and related by
, then and are identically distributed, since
When and are identically distributed, we write
.
- Again, suppose
and correspond to and , two distribution functions of the same type related by
. Then it is easy to see that
iff
. In fact, if the expectation exists for one, then
. Furthermore, is finite iff is. And in this case,
. In general, convergence of moments is a “typical” property.
- We can partition the set of distribution functions into disjoint subsets of functions belonging to the same types, since the binary relation
is an equivalence relation.
- 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 ).
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"type of a distribution function" is owned by CWoo.
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(view preamble)
| Other names: |
centering factor, scale parameter, location parameter |
| Also defines: |
type, scale factor, location factor, standard distribution function, location family, scale family |
This object's parent.
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Cross-references: normal distribution, variance, equivalence class, probability space, fixed, real, equivalence relation, binary relation, functions, subsets, disjoint, partition, property, moments, finite, expectation, iff, easy to see, identically distributed, random variables, units, right, graph, parameter, distribution functions
There are 38 references to this entry.
This is version 10 of type of a distribution function, born on 2006-11-22, modified 2006-11-29.
Object id is 8582, canonical name is TypeOfADistributionFunction.
Accessed 4361 times total.
Classification:
| AMS MSC: | 60E05 (Probability theory and stochastic processes :: Distribution theory :: Distributions: general theory) | | | 62E10 (Statistics :: Distribution theory :: Characterization and structure theory) |
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Pending Errata and Addenda
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