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. 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. 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 : This family is obtained by adding a constant to a random variable. Let X {displaystyle X} be a random variable and a ∈ R {displaystyle ain R} be a constant. Let Y = X + a { extstyle Y=X+a} . Then This family is obtained by multiplying a random variable with a constant. Let X {displaystyle X} be a random variable and c ∈ R + {displaystyle cin R^{+}} be a constant. Let Y = c X { extstyle Y=cX} . Then This family is obtained by multiplying a random variable with a constant and then adding some other constant to it. Let X {displaystyle X} be a random variable , a ∈ R {displaystyle ain R} and c ∈ R + {displaystyle cin R^{+}} be constants. Let Y = c X + a {displaystyle Y=cX+a} . Then Note that it is important that a ∈ R { extstyle ain R} and c ∈ R + {displaystyle cin R^{+}} in order to satisfy the properties mentioned in the following section. The transformation applied to the random variable must satisfy the following properties.