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 tilting
Thus, formula_5 has density formula_12. The set of formula_13 parameterized distributions is known as the natural exponential family generated by formula_1. The exponential tilting of a random vector formula_1 has an analogous definition: formula_16, where formula_17. This “exponentially tilted measure" is a probability distribution that in many cases has the same parametric form as that of formula_1, so that variates can easily be generated. However, for some distributions, the exponentially tilted distribution does not 
belong to the same parametric family as formula_3; an example is the Pareto distribution with formula_20, where formula_21 is well defined for formula_22 but is not a standard distribution. In such examples, r.v. generation may not always be straightforward.