# Parametric family

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.