# Parametric model

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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 $mathcal\left\{P\right\} = \left\{ f_\left\{ heta\right\} mid heta in Theta \right\}$ where $Theta$ is the set of all possible parameters.

Examples

* For each real number &mu; and each positive number &sigma;2 there is a normal distribution whose expected value is &mu; and whose variance is &sigma;2. Its probability density function is

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

Thus the family of normal distributions is parametrized by the pair (&mu;, &sigma;2).

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

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

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

Thus the family of Poisson distributions is parametrized by the positive number &lambda;.

The family of Poisson distributions is an exponential family.

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