Exponential family
What is important to note, and what characterizes all exponential family variants, is that the parameter(s) and the observation variable(s) must factorize (can be separated into products each of which involves only one type of variable), either directly or within either part (the base or exponent) of an exponentiation operation. Generally, this means that all of the factors constituting the density or mass function must be of one of the following forms:
Exponential dispersion model
In probability and statistics, the class of exponential dispersion models (EDM) is a set of probability distributions that represents a generalisation of the natural exponential family. 
Exponential dispersion models play an important role in statistical theory, in particular in generalized linear models because they have a special structure which enables deductions to be made about appropriate statistical inference. 
There are two versions to formulate an exponential disperson model.
In the univariate case, a real-valued random variable formula_1 belongs to the additive exponential dispersion model with canonical parameter formula_2 and index parameter formula_3, formula_4, if its probability density function can be written as