# Parametric statistics

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Parametric inferential statistical methods are mathematical procedures for statistical hypothesis testing which assume that the distributions of the variables being assessed belong to known parametrized families of probability distributions. In that case we speak of 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

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

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.

For example, analysis of variance assumes that the underlying distributions are normally distributed and that the variances of the distributions being compared are similar. The Pearson product-moment correlation coefficient assumes normality.

While parametric techniques are robust – that is, they often retain considerable power to detect differences or similarities even when these assumptions are violated – some distributions violate the assumptions so markedly that a non-parametric alternative is more likely to detect a difference or similarity.