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 function
where is a positive real number, and in which the argument occurs as an exponent. For real numbers and a function of the form formula_1 is also an exponential function, as it can be rewritten as