Exponential family of distributions | Vose Software

# 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.