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.
Types of group families
For a fixed distribution , as varies from to , the distributions that we obtain constitute the location family.
Location - Scale Family
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.
- Closure under composition
- Closure under inversion