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

 * 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


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

Thus the family of normal distributions is parametrized by the pair (&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


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

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

The family of Poisson distributions is an exponential family.