# Parametric model

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.

## Examples

${\displaystyle \varphi _{\mu ,\sigma ^{2}}(x)={1 \over \sigma }\cdot {1 \over {\sqrt {2\pi }}}\exp \left({-1 \over 2}\left({x-\mu \over \sigma }\right)^{2}\right)}$

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

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

The family of Poisson distributions is an exponential family.