- 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 aparametrized family ofprobability distribution s, 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.

**Examples*** For each

real number μ and each positive number σ^{2}there is anormal distribution whoseexpected value is μ and whosevariance is σ^{2}. 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, σ

^{2}).This parametrized family is both an

exponential family and alocation-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 distribution s is parametrized by the positive number λ.The family of Poisson distributions is an

exponential family .

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