Inprobability theory, especially in mathematicalstatistics, alocation–scale family is afamily ofprobability distributions parametrized by alocation parameter and a non-negativescale parameter. For anyrandom variable${\displaystyle X}$ whose probability distribution function belongs to such a family, the distribution function of${\displaystyle Y\stackrel{d}{=}a+bX}$ also belongs to the family (where${\displaystyle \stackrel{d}{=}}$ means "equal in distribution"—that is, "has the same distribution as").

In other words, a class${\displaystyle \mathrm{\Omega }}$ of probability distributions is a location–scale family if for allcumulative distribution functions${\displaystyle F\in \mathrm{\Omega }}$ and any real numbers${\displaystyle a\in \mathbb{R}}$ and${\displaystyle b>0}$, the distribution function${\displaystyle G(x)=F(a+bx)}$ is also a member of${\displaystyle \mathrm{\Omega }}$.

• If${\displaystyle X}$ has acumulative distribution function${\displaystyle F_X(x)=P(X\le x)}$, then${\displaystyle Y=a+bX}$ has a cumulative distribution function${\displaystyle F_Y(y)=F_X\left(\frac{y-a}{b}\right)}$.
• If${\displaystyle X}$ is adiscrete random variable withprobability mass function${\displaystyle p_X(x)=P(X=x)}$, then${\displaystyle Y=a+bX}$ is a discrete random variable with probability mass function${\displaystyle p_Y(y)=p_X\left(\frac{y-a}{b}\right)}$.
• If${\displaystyle X}$ is acontinuous random variable withprobability density function${\displaystyle f_X(x)}$, then${\displaystyle Y=a+bX}$ is a continuous random variable with probability density function${\displaystyle f_Y(y)=\frac{1}{b}{f}_X\left(\frac{y-a}{b}\right)}$.
• If${\displaystyle X}$ has amoment generating function${\displaystyle M_X(t)}$, then${\displaystyle Y=a+bX}$ has a moment generating function${\displaystyle M_Y(t)=e^{at}{M}_X(bt)}$.
• If${\displaystyle X}$ has acharacteristic function${\displaystyle {\phi }_X(t)}$, then${\displaystyle Y=a+bX}$ has a characteristic function${\displaystyle {\phi }_Y(t)=e^{iat}{\phi }_X(bt)}$.

Moreover, if${\displaystyle X}$ and${\displaystyle Y}$ are two random variables whose distribution functions are members of the family, and assuming existence of the first two moments and${\displaystyle X}$ has zero mean and unit variance,then${\displaystyle Y}$ can be written as${\displaystyle Y\stackrel{d}{=}{\mu }_Y+{\sigma }_{Y}X}$ , where${\displaystyle {\mu }_Y}$ and${\displaystyle {\sigma }_Y}$ are the mean and standard deviation of${\displaystyle Y}$.

Indecision 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 atwo-moment decision model can apply, and decision-making can be framed in terms of themeans and thevariances of the distributions.^[1][2][3]

Often, location–scale families are restricted to those where all members have the same functional form. Most location–scale families areunivariate, though not all. Well-known families in which the functional form of the distribution is consistent throughout the family include the following:

• Normal distribution
• Elliptical distributions
• Cauchy distribution
• Uniform distribution (continuous)
• Uniform distribution (discrete)
• Logistic distribution
• Laplace distribution
• Student's t-distribution
• Generalized extreme value distribution

The following shows how to implement a location–scale family in a statistical package or programming environment where only functions for the "standard" version of a distribution are available. It is designed forR but should generalize to any language and library.

The example here is of theStudent'st-distribution, which is normally provided in R only in its standard form, with a singledegrees of freedom parameterdf. The versions below with_ls appended show how to generalize this to ageneralized Student's t-distribution with an arbitrary location parameterm and scale parameters.

Note that these generalized functions do not have standard deviations since standardt distributions do not have a standard deviation of 1.

• http://www.randomservices.org/random/special/LocationScale.html

• Parametric statistics
• Types of probability distributions
• Location-scale family probability distributions

• Articles with short description
• Short description matches Wikidata