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 factorial
The exponential factorial of a positive integer "n", denoted by "n"$, is "n" raised to the power of "n" − 1, which in turn is raised to the power of "n" − 2, and so on and so forth, that is,