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<b> [[Exponential family|Exponential family]]</B>: If a *[[random variable]] X has a | <b> [[Exponential family|Exponential family]]</B>: If a *[[random variable]] X has a [[probability distribution]] (*discrete case) or a [[probability density function]] [*continuous case) that can be written in the form | ||
:<math>exp\{a(x)b(\theta) + c(\theta) + d(x)\}</math> | :<math>exp\{a(x)b(\theta) + c(\theta) + d(x)\}</math> | ||
where <math>\theta</math> is a parameter and a, b, c, and d are known functions, then it is a member of the exponential family. The *[[Poisson distribution]], the *[[binomial distribution]], and the *[[Normal Distribution|normal distribution]] are members of this family. If <math>a(x) = x</math>, then the distribution is in canonical form and <math>b(\theta)</math> is a natural parameter of the distribution. | where <math>\theta</math> is a parameter and a, b, c, and d are known functions, then it is a member of the exponential family. The *[[Poisson distribution]], the *[[binomial distribution]], and the *[[Normal Distribution|normal distribution]] are members of this family. If <math>a(x) = x</math>, then the distribution is in canonical form and <math>b(\theta)</math> is a natural parameter of the distribution. |