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 :

This family is obtained by adding a constant to a random variable. Let ${\displaystyle X}$  be a random variable and ${\displaystyle a\in R}$  be a constant. Let ${\textstyle Y=X+a}$  . Then $${\displaystyle F_{Y}(y)=P(Y\leq y)=P(X+a\leq y)=P(X\leq y-a)=F_{X}(y-a)}$$ For a fixed distribution, as ${\displaystyle a}$  varies from ${\displaystyle -\infty }$  to ${\displaystyle \infty }$ , the distributions that we obtain constitute the location family.

This family is obtained by multiplying a random variable with a constant. Let ${\displaystyle X}$  be a random variable and ${\displaystyle c\in R^{+}}$  be a constant. Let ${\textstyle Y=cX}$  . Then$${\displaystyle F_{Y}(y)=P(Y\leq y)=P(cX\leq y)=P(X\leq y/c)=F_{X}(y/c)}$$ 

This family is obtained by multiplying a random variable with a constant and then adding some other constant to it. Let ${\displaystyle X}$  be a random variable, ${\displaystyle a\in R}$  and ${\displaystyle c\in R^{+}}$ be constants. Let ${\displaystyle Y=cX+a}$ . Then

$${\displaystyle F_{Y}(y)=P(Y\leq y)=P(cX+a\leq y)=P(X\leq (y-a)/c)=F_{X}((y-a)/c)}$$ 

Note that it is important that ${\textstyle a\in R}$  and ${\displaystyle c\in R^{+}}$  in order to satisfy the properties mentioned in the following section.

The transformation applied to the random variable must satisfy the properties of closure under composition and inversion.^[1]