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(n, p):

 =

 = x

 =

 = nln(1-p)

Geometric(p):

 = ln[1-p]

 = x

 = 0

 = ln[p]

Poisson(l):

 = ln[l]

 = x

 = -ln[x!]

 = -l

Gamma(a, b):

 = -1/b 

 = x

 =

 = a ln[1/b]

Normal(m, 1):

 = m 

 = x

 =

 = - ½m2

Inverse Gaussian(m, l):

 = m-2 

 = x

 =

 =

Rayleigh(b):

 =

 = x2 

 = ln[x]

 = -2ln[b]

 

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

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

 

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