Parametric model

A parametric model is a set of related mathematical equations in which alternative scenarios are defined by changing the assumed values of a set of fixed coefficients (parameters). In statistics, a parametric model is a parametrized family of probability distributions, one of which is presumed to describe the way a population is distributed.

Formally, a parametric model is the set $mathcal\{P\}\; =\; \{\; f\_\{\; heta\}\; mid\; heta\; in\; Theta\; \}$ where $Theta$ is the set of all possible parameters.

Examples

* For each real number μ and each positive number σ² there is a normal distribution whose expected value is μ and whose variance is σ². Its probability density function is

::$varphi\_\{mu,sigma^2\}(x)\; =\; \{1\; over\; sigma\}cdot\{1\; over\; sqrt\{2pi\; expleft(\; \{-1\; over\; 2\}\; left(\{x\; -\; mu\; over\; sigma\}\; ight)^2\; ight)$

Thus the family of normal distributions is parametrized by the pair (μ, σ²).

This parametrized family is both an exponential family and a location-scale family

* For each positive real number λ there is a Poisson distribution whose expected value is λ. Its probability mass function is

::$f(x)\; =\; \{lambda^x\; e^\{-lambda\}\; over\; x!\}\; mathrm\{for\}\; xin\{,0,1,2,3,dots,\}.$

Thus the family of Poisson distributions is parametrized by the positive number λ.

The family of Poisson distributions is an exponential family.