parametric family
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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 differences depend only on the chosen values for a set of parameters.{{Cite journal |date=2006 |title=All of Nonparametric Statistics |url=https://link.springer.com/book/10.1007/0-387-30623-4 |journal=Springer Texts in Statistics |language=en |doi=10.1007/0-387-30623-4|isbn=978-0-387-25145-5 }}
Common examples are parametrized (families of) functions, probability distributions, curves, shapes, etc.{{Citation needed|date=August 2021}}
In probability and its applications
{{main|Statistical model}}
File:Probability distribution functions for normal distribution.svg (from the same parametric family).]]
For example, the probability density function {{math|fX}} of a random variable {{mvar|X}} may depend on a parameter {{mvar|θ}}. In that case, the function may be denoted to indicate the dependence on the parameter {{mvar|θ}}. {{mvar|θ}} 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 {{math|Θ}} denotes the parameter space, the set of all possible values that the parameter {{mvar|θ}} can take. As an example, the normal distribution is a family of similarly-shaped distributions parametrized by their mean and their variance.{{Cite book|last=Mukhopadhyay|first=Nitis|title=Probability and Statistical Inference|publisher=Marcel Dekker, Inc.|year=2000|isbn=0-8247-0379-0|location=United States of America|pages=282–283; 341}}{{Cite web|title=Parameter of a distribution|url=https://www.statlect.com/glossary/parameter|access-date=2021-08-04|website=www.statlect.com}}
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.{{Citation needed|date=August 2021}}
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.{{Citation needed|date=August 2021}}
File:Quadratic equation coefficients.png, varying each of the three coefficients independently.]]
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.{{Citation needed|date=August 2021}}