Parametric family explained

In mathematics and its applications, a parametric family or a parameterized family is a family of objects (a set of related objects) whose differences depend only on the chosen values for a set of parameters.^[1]

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

In probability and its applications

See main article: Statistical model. For example, the probability density function of a random variable may depend on a parameter . In that case, the function may be denoted

f _{X(} ⋅ ; \theta) 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 \{ f _{X(} ⋅ ; \theta) \mid \theta \in \Theta \} , where denotes the parameter space, the set of all possible values that the parameter can take. As an example, the normal distribution is a family of similarly-shaped distributions parametrized by their mean and their variance.^{[2] [3]}

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

Notes and References

1. 2006 . All of Nonparametric Statistics . Springer Texts in Statistics . en . 10.1007/0-387-30623-4.
2. Book: Mukhopadhyay, Nitis. Probability and Statistical Inference. Marcel Dekker, Inc.. 2000. 0-8247-0379-0. United States of America. 282–283; 341.
3. Web site: Parameter of a distribution. 2021-08-04. www.statlect.com.