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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.
In probability and its applications
For example, the probability density function of a random variable X may depend on a parameter . In that case, the function may be denoted to indicate the dependence on the parameter . 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 denotes the set of all possible values that the parameter can take. As an example, the normal distribution is a family of similarly-shaped distributions parametrized by their mean and their variance.