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# Exponential family of distributions

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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