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

## Examples

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

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

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

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

The family of Poisson distributions is an exponential family.

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