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 differences depend only on the chosen values for 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 $θ$. In that case, the function may be denoted $$ f_X( \cdot \, ; \theta) $$ 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 $$ \{ f_X( \cdot \, ; \theta) \mid \theta \in \Theta \} $$, 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.

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.