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 and a non-negative scale parameter; if X is any random variable whose probability distribution belongs to such a family, then Y =d (a + bX) is another (where =d means "is equal in distribution to" — that is, "has the same distribution as"), and every distribution in the family is of that form. Moreover, if X is a zero-mean, unit-variance member of the family, then every member Y of the family can be written as Y =d (μY + σYX), where μY and σY are the mean and standard deviation of Y.
In other words, a class Ω of probability distributions is a location-scale family if whenever F is the cumulative distribution function of a member of Ω and a is any real number and b > 0, then G(x) = F(a + bx) is also the cumulative distribution function of a member of Ω.
In decision theory, if all alternative distributions available to a decision-maker are in the same location-scale family, then a two-moment decision model applies, and decision-making can be framed in terms of the means and the variances of the distributions.
Read more about Location-scale Family: Examples, Converting A Single Distribution To A Location-scale Family
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