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. Consideration of a particular family of distributions as a group family can, in statistical theory, lead to the identification of an ancillary statistic. Types of group families. 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 Family. This family is obtained by adding a constant to a random variable. Let formula_1 be a random variable and formula_2 be a constant. Let formula_3 . Then formula_4For a fixed distribution , as formula_5 varies from formula_6 to formula_7 , the distributions that we obtain constitute the location family. Scale Family. This family is obtained by multiplying a random variable with a constant. Let formula_1 be a random variable and formula_9 be a constant. Let formula_10 . Thenformula_11 Location - Scale Family. This family is obtained by multiplying a random variable with a  constant and then adding some other constant to it. Let formula_12 be a random variable , formula_2 and formula_9be constants. Let formula_15. Then formula_16 Note that it is important that formula_17 and formula_18 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.