# Location–scale family

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 $\displaystyle{ X }$ whose probability distribution function belongs to such a family, the distribution function of $\displaystyle{ Y \stackrel{d}{=} a + b X }$ 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 $\displaystyle{ \Omega }$ of probability distributions is a location–scale family if for all cumulative distribution functions $\displaystyle{ F \in \Omega }$ and any real numbers $\displaystyle{ a \in \mathbb{R} }$ and $\displaystyle{ b \gt 0 }$, the distribution function $\displaystyle{ G(x) = F(a + b x) }$ is also a member of $\displaystyle{ \Omega }$.

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

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 $\displaystyle{ Y }$.

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.

## Examples

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.