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In mathematics and its applications, a parametric family or a parameterized family is a family of objects (a set of related objects) whose definitions depend on a set of parameters.

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

In probability and its applications

For example, the probability density function \( f_X \) of a random variable X may depend on a parameter \theta . In that case, the function may be denoted\( f_X( \cdot \, ; \theta) \) to indicate the dependence on the parameter \( \theta . \theta \) 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 \( \{ f_X( \cdot \, ; \theta) \mid \theta \in \Theta \} \) , where \( \Theta \) denotes the set of all possible values that the parameter \( \theta \) can take. In particular the normal distribution is actually a family of similarly shaped distributions parametrized by their mean 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, the Cobb-Douglas production function is a family of production functions parametrized by the elasticities of output with respect to the various factors of production.

In algebra, the quadratic equation, 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

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

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