Location–scale family

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Short description: Family of probability distributions

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 [math]\displaystyle{ X }[/math] whose probability distribution function belongs to such a family, the distribution function of [math]\displaystyle{ Y \stackrel{d}{=} a + b X }[/math] also belongs to the family (where [math]\displaystyle{ \stackrel{d}{=} }[/math] means "equal in distribution"—that is, "has the same distribution as").

In other words, a class [math]\displaystyle{ \Omega }[/math] of probability distributions is a location–scale family if for all cumulative distribution functions [math]\displaystyle{ F \in \Omega }[/math] and any real numbers [math]\displaystyle{ a \in \mathbb{R} }[/math] and [math]\displaystyle{ b \gt 0 }[/math], the distribution function [math]\displaystyle{ G(x) = F(a + b x) }[/math] is also a member of [math]\displaystyle{ \Omega }[/math].

  • If [math]\displaystyle{ X }[/math] has a cumulative distribution function [math]\displaystyle{ F_X(x)= P(X\le x) }[/math], then [math]\displaystyle{ Y{=} a + b X }[/math] has a cumulative distribution function [math]\displaystyle{ F_Y(y) = F_X\left(\frac{y-a}{b}\right) }[/math].
  • If [math]\displaystyle{ X }[/math] is a discrete random variable with probability mass function [math]\displaystyle{ p_X(x)= P(X=x) }[/math], then [math]\displaystyle{ Y{=} a + b X }[/math] is a discrete random variable with probability mass function [math]\displaystyle{ p_Y(y) = p_X\left(\frac{y-a}{b}\right) }[/math].
  • If [math]\displaystyle{ X }[/math] is a continuous random variable with probability density function [math]\displaystyle{ f_X(x) }[/math], then [math]\displaystyle{ Y{=} a + b X }[/math] is a continuous random variable with probability density function [math]\displaystyle{ f_Y(y) = \frac{1}{b}f_X\left(\frac{y-a}{b}\right) }[/math].

Moreover, if [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are two random variables whose distribution functions are members of the family, and assuming existence of the first two moments and [math]\displaystyle{ X }[/math] has zero mean and unit variance, then [math]\displaystyle{ Y }[/math] can be written as [math]\displaystyle{ Y \stackrel{d}{=} \mu_Y + \sigma_Y X }[/math] , where [math]\displaystyle{ \mu_Y }[/math] and [math]\displaystyle{ \sigma_Y }[/math] are the mean and standard deviation of [math]\displaystyle{ Y }[/math].

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.[1][2][3]


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

Converting a single distribution to a location–scale family

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 for R but should generalize to any language and library.

The example here is of the Student's t-distribution, which is normally provided in R only in its standard form, with a single degrees of freedom parameter df. The versions below with _ls appended show how to generalize this to a generalized Student's t-distribution with an arbitrary location parameter mu and scale parameter sigma.

Probability density function (PDF): dt_ls(x, df, mu, sigma) = 1/sigma * dt((x - mu)/sigma, df)
Cumulative distribution function (CDF): pt_ls(x, df, mu, sigma) = pt((x - mu)/sigma, df)
Quantile function (inverse CDF): qt_ls(prob, df, mu, sigma) = qt(prob, df)*sigma + mu
Generate a random variate: rt_ls(df, mu, sigma) = rt(df)*sigma + mu

Note that the generalized functions do not have standard deviation sigma since the standard t distribution does not have standard deviation of 1.


  1. Meyer, Jack (1987). "Two-Moment Decision Models and Expected Utility Maximization". American Economic Review 77 (3): 421–430. 
  2. Mayshar, J. (1978). "A Note on Feldstein's Criticism of Mean-Variance Analysis". Review of Economic Studies 45 (1): 197–199. 
  3. Sinn, H.-W. (1983). Economic Decisions under Uncertainty (Second English ed.). North-Holland. 

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