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 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   (μ_Y + σ_YX), where μ_Y and σ_Y 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.

Examples

In general, any random variable X is a member of a location-scale family with an infinite number of members; each of these members Y is distributed the same as a + bX for real numbers a and b > 0. However, the form of the distribution function for a + bX, and hence for Y, may differ from the form of the distribution function for X. Well-known families in which the functional form of the distribution is retained throughout the family include the following:

• Normal distribution
• Elliptical distribution
• Cauchy distribution
• Uniform distribution (continuous)
• Uniform distribution (discrete)
• Logistic distribution
• Laplace distribution

Further references

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