Parametric model

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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.

[edit] Examples

\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
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.

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