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 fX of a random variable X may depend on a parameter θ. In that case, the function may be denoted f X ( ⋅ ; θ ) {\displaystyle 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 ( ⋅ ; θ ) ∣ θ ∈ Θ } {\displaystyle \{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.
See also Indexed family
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