Parametric family

Inmathematicsand its applications, aparametric familyor aparameterized familyis afamilyof objects (a set of related objects) whose differences depend only on the chosen values for a set ofparameters.

Common examples are parametrized (families of)functions,probability distributions,curves, shapes, etc.

In probability and its applications

Main article:Statistical model

A graph of the probability density functions of severalnormal distributions(from the same parametric family).

For example, the probability density functionf_Xof a random variableXmay depend on a parameterθ.In that case, the function may be denoted$f_{X}(\cdot \,;\theta )$to indicate the dependence on the parameterθ.θis 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 theparametric familyof densities is the set of functions$\{f_{X}(\cdot \,;\theta )\mid \theta \in \Theta \}$,whereΘdenotes theparameter space,the set of all possible values that the parameterθcan take. As an example, thenormal distributionis a family of similarly-shaped distributions parametrized by theirmeanand theirvariance.

Indecision theory,two-moment decision modelscan be applied when the decision-maker is faced with random variables drawn from alocation-scale familyof probability distributions.

In algebra and its applications

A three-dimensional graph of aCobb–Douglas production function.

Ineconomics,theCobb–Douglas production functionis a family ofproduction functionsparametrized by theelasticitiesof output with respect to the variousfactors of production.

Graphs of severalquadratic polynomials,varying each of the three coefficients independently.

Inalgebra,thequadratic equation,for example, is actually a family of equations parametrized by thecoefficientsof the variable and of its square and by theconstant term.

See also

• Indexed family