 |
Download a pdf copy of this help file here |
|
Download a pdf copy of this help file here |
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.
Categorising probability distributions this way is useful in Extreme Value Theory.
