In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and its applications, a parametric family or a parameterized family is a
family
Family (from la, familia) is a group of people related either by consanguinity (by recognized birth) or affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its members and of society. Idea ...
of objects (a set of related objects) whose differences depend only on the chosen values for a set of
parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
s.
Common examples are parametrized (families of)
functions,
probability distributions, curves, shapes, etc.
In probability and its applications
For example, the probability density function of a random variable may depend on a parameter . In that case, the function may be denoted
to indicate the dependence on the parameter . is not a formal argument of the function as it is considered to be fixed. However, each different value of the parameter gives a different probability density function. Then the ''parametric family'' of densities is the set of functions
, where denotes the
parameter space The parameter space is the space of possible parameter values that define a particular mathematical model, often a subset of finite-dimensional Euclidean space. Often the parameters are inputs of a function, in which case the technical term for ...
, the set of all possible values that the parameter can take. As an example, the
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
is a family of similarly-shaped distributions parametrized by their
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set.
For a data set, the '' ar ...
and their
variance.
In
decision theory,
two-moment decision models can be applied when the decision-maker is faced with random variables drawn from a
location-scale family of probability distributions.
In algebra and its applications
In
economics
Economics () is the social science that studies the production, distribution, and consumption of goods and services.
Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics anal ...
, the
Cobb–Douglas production function
In economics and econometrics, the Cobb–Douglas production function is a particular functional form of the production function, widely used to represent the technological relationship between the amounts of two or more inputs (particularly p ...
is a family of
production functions parametrized by the
elasticities of output with respect to the various
factors of production.
In
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
, the
quadratic equation
In algebra, a quadratic equation () is any equation that can be rearranged in standard form as
ax^2 + bx + c = 0\,,
where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not qu ...
, for example, is actually a family of equations parametrized by the
coefficients of the variable and of its square and by the
constant term.
See also
*
Indexed family
References
{{reflist
Mathematical terminology
Theory of probability distributions