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One often sees reference to the exponential family of distributions in probability theory texts. This refers to a group of distributions whose probability density or mass function is of the general form:

 

f(x) = exp[A(q)B(x) +C(x) + D(q)]

 

where A, B, C and D are functions and q is a uni-dimensional or multidimensional parameter.

 

Examples of distributions in the exponential family are: Binomial, Geometric, Poisson, Gamma, Normal, Inverse Gaussian and Rayleigh. For these distributions:

 

Distribution

A(q)

B(x)

C(x)

D(q)

Binomial(p, n):

 = {ln\Big[\frac{p}{1-p}\Big]}

 = x

 = {ln\Big[\binom{n}{x}\Big]}

 = nln(1-p)

Geometric(p)-1:

 = ln[1-p]

 = x

 = 0

 = ln[p]

Poisson(l):

 = ln[l]

 = x

 = -ln[x!]

 = -l

Gamma(0,b,a):

 = -1/b 

 = x

 = {ln=\Big[\frac{x^{\alpha -1}}{\Gamma(\alpha)}\Big]}

 = a ln[1/b]

Normal(m, 1):

 = m 

 = x

 = {-\Big(\frac{x^{2}+ln[2\pi ]}{2}\Big)}

 = - ½m2

Inverse Gaussian(m, l):

 = m-2 

 = x

 = {-\frac{1}{2}\Big(ln\Big[\frac{2\pi x^{3}}{\lambda}\Big]+\frac{\lambda}{x}\Big)}

 = {\frac{-1}{\sqrt{-2\mu}}}

Rayleigh(b):

 = {-\frac{1}{2b^{2}}}

 = x2 

 = ln[x]

 = -2ln[b]

 

Those distributions with B(x) = x form a group known as the natural exponential family.

 

Categorising probability distributions this way is useful in Extreme Value Theory.

 


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