# Location-scale family

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Location-scale family

In probability theory, especially as that field is used in statistics, a location-scale family is a family of univariate probability distributions parametrized by a location parameter &mu; and a scale parameter &sigma; &ge; 0; if "X" is any random variable whose probability distribution belongs to such a family, then "Y" = &mu; + &sigma;"X" is another, and every distribution in the family is of that form.

In other words, a class &Omega; of probability distributions is a location-scale family if whenever "F" is the cumulative distribution function of a member of &Omega; and &mu; is any real number and &sigma; > 0, then "G"("x") = "F"(&mu; + &sigma;"x") is also the cumulative distribution function of a member of &Omega;.

Examples

* normal distribution
* Cauchy distribution
* uniform distribution
* logistic distribution

References

http://www.ds.unifi.it/VL/VL_EN/special/special1.html

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