In probability theory, especially as it is used in statistics, a group family of probability distributions is one obtained by subjecting a random variable with a fixed distribution to a suitable transformation, such as a location–scale family, or otherwise one of probability distributions acted upon by a group. Considering a family of distributions as a group family can, in statistical theory, lead to identifying ancillary statistics.

Types

A group family can be generated by subjecting a random variable with a fixed distribution to some suitable transformations. Different types of group families are as follows :

Location

This family is obtained by adding a constant to a random variable. Let X be a random variable and a \in R be a constant. Let Y = X + a. Then For a fixed distribution, as a varies from -\infty to \infty, the distributions that we obtain constitute the location family.

Scale

This family is obtained by multiplying a random variable with a constant. Let X be a random variable and c \in R^+ be a constant. Let Y = cX. Then

Location–scale

This family is obtained by multiplying a random variable with a constant and then adding some other constant to it. Let X be a random variable, a \in R and c \in R^+be constants. Let Y = cX + a. Then Note that it is important that a \in R and c \in R^+ in order to satisfy the properties mentioned in the following section.

Transformation

The transformation applied to the random variable must satisfy the properties of closure under composition and inversion.