# Group family

In probability theory, especially as that field is used in statistics, a **group family** of probability distributions is a family obtained by subjecting a random variable with a fixed distribution to a suitable family of transformations such as a location-scale family, or otherwise a family of probability distributions acted upon by a group.[1]

Consideration of a particular family of distributions as a group family can, in statistical theory, lead to the identification of an ancillary statistic.[2]

## Types of group families

A group family can be generated by subjecting a random variable with a fixed distribution to some suitable transformations.[1] Different types of group families are as follows :

### Location Family

This family is obtained by adding a constant to a random variable. Let be a random variable and be a constant. Let . Then

For a fixed distribution , as varies from to , the distributions that we obtain constitute the location family.

### Scale Family

This family is obtained by multiplying a random variable with a constant. Let be a random variable and be a constant. Let . Then

### Location - Scale Family

This family is obtained by multiplying a random variable with a constant and then adding some other constant to it. Let be a random variable , and be constants. Let . Then

Note that it is important that and in order to satisfy the properties mentioned in the following section.

## Properties of the transformations

The transformation applied to the random variable must satisfy the following properties.[1]

- Closure under composition
- Closure under inversion