Parametric model

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.


* 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.

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