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In probability theory, especially in mathematical statistics, a location–scale family is a family of probability distributions parametrized by a location parameter and a non-negative scale parameter. For any random variable X {\displaystyle X} whose probability distribution function belongs to such a family, the distribution function of Y = d a + b X {\displaystyle Y{\stackrel {d}{=}}a+bX} also belongs to the family.
In other words, a class Ω {\displaystyle \Omega } of probability distributions is a location–scale family if for all cumulative distribution functions F ∈ Ω {\displaystyle F\in \Omega } and any real numbers a ∈ R {\displaystyle a\in \mathbb {R} } and b > 0 {\displaystyle b>0} , the distribution function G = F {\displaystyle G=F} is also a member of Ω {\displaystyle \Omega } .
Moreover, if X {\displaystyle X} and Y {\displaystyle Y} are two random variables whose distribution functions are members of the family, and assuming existence of the first two moments and X {\displaystyle X} has zero mean and unit variance, then Y {\displaystyle Y} can be written as Y = d μ Y + σ Y X {\displaystyle Y{\stackrel {d}{=}}\mu _{Y}+\sigma _{Y}X} , where μ Y {\displaystyle \mu _{Y}} and σ Y {\displaystyle \sigma _{Y}} are the mean and standard deviation of Y {\displaystyle Y} .
In decision theory, if all alternative distributions available to a decision-maker are in the same location–scale family, and the first two moments are finite, then a two-moment decision model can apply, and decision-making can be framed in terms of the means and the variances of the distributions.