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title: "Exponential family of distributions"
canonical: "https://modelassist.epixanalytics.com/space/EA/26576116/Exponential%20family%20of%20distributions"
format: markdown
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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](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575204/), [Geometric](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575207/), [Poisson](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575214/), [Gamma](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575231/), [Normal](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575246/), [Inverse Gaussian](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575236/) and [Rayleigh](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575252/). For these distributions:

 

| ***Distribution*** | ***A(q)*** | ***B(x)*** | ***C(x)*** | ***D(q)*** |
| --- | --- | --- | --- | --- |
| Binomial(p, *n*): | * = *> Macro (mathinline) | * = x* | * = *> Macro (mathinline) | * = n*ln(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* | * = *> Macro (mathinline) | * =* *a* ln[*1/b*] |
| Normal(*m*, 1): | * =* *m * | * = x* | * = *> Macro (mathinline) | * = -* ½*m*2 |
| Inverse Gaussian(*m, l*): | * =* *m*-2* * | * = x* | * = *> Macro (mathinline) | * = *> Macro (mathinline) |
| Rayleigh(*b*): | * = *> Macro (mathinline) | * = x*2* * | * =* 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](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575228/).

 

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