probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

, especially as that field is used in

statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...

, a group family of

probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...

s 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 A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

. Consideration of a particular family of distributions as a group family can, in

statistical theory The theory of statistics provides a basis for the whole range of techniques, in both study design and data analysis, that are used within applications of statistics. The theory covers approaches to statistical-decision problems and to statistica ...

, lead to the identification of an

ancillary statistic An ancillary statistic is a measure of a sample whose distribution (or whose pmf or pdf) does not depend on the parameters of the model. An ancillary statistic is a pivotal quantity that is also a statistic. Ancillary statistics can be used to c ...

.Cox, D.R. (2006) ''Principles of Statistical Inference'', CUP. (Section 4.4.2)

Types of group families

A group family can be generated by subjecting a

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...

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

. Let

X

be a

and

a \in R

be a constant. Let

Y = X + a

. Then

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

a

varies from

-\infty

\infty

, the distributions that we obtain constitute the location family.

Scale Family

This family is obtained by multiplying a

with a constant. Let

X

be a

and

c \in R^+

be a constant. Let

Y = cX

. Then

F_Y(y) = P(Y\leq y) = P(cX \leq y) = P(X \leq y/c) = F_X(y/c)

Location - Scale Family

This family is obtained by multiplying a

with a constant and then adding some other constant to it. Let

X

be a

a \in R

and

c \in R^+

be constants. Let

Y = cX + a

. Then

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

a \in R

and

c \in R^+

in order to satisfy the properties mentioned in the following section.

Properties of the transformations

The

transformation Transformation may refer to: Science and mathematics In biology and medicine * Metamorphosis, the biological process of changing physical form after birth or hatching * Malignant transformation, the process of cells becoming cancerous * Trans ...

applied to the random variable must satisfy the following properties. * Closure under composition * Closure under inversion

References

Parametric statistics Types of probability distributions {{Statistics-stub