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.[citation needed]
Common examples are parametrized (families of) functions, probability distributions, curves, shapes, etc.[citation needed]
In probability and its applications
![A graph of several normal distributions.](http://upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Probability_distribution_functions_for_normal_distribution.svg/220px-Probability_distribution_functions_for_normal_distribution.svg.png)
For example, the probability density function fX of a random variable X may depend on a parameter θ. In that case, the function may be denoted to indicate the dependence on the parameter θ. 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 Θ denotes the parameter space, the set of all possible values that the parameter θ can take. As an example, the normal distribution is a family of similarly-shaped distributions parametrized by their mean and their variance.[1][2]
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]
In algebra and its applications
![](http://upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Cobb_douglas.png/220px-Cobb_douglas.png)
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]
![Graphs of several quadratic equations](http://upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Quadratic_equation_coefficients.png/220px-Quadratic_equation_coefficients.png)
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]
See also
References
- ^ Mukhopadhyay, Nitis (2000). Probability and Statistical Inference. United States of America: Marcel Dekker, Inc. pp. 282–283, 341. ISBN 0-8247-0379-0.
- ^ "Parameter of a distribution". www.statlect.com. Retrieved 2021-08-04.