Exponentiation
Exponentiation is a mathematical operation, written as , involving two numbers, the "base" and the "exponent" . When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, is the product of multiplying bases:
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.