Exponentiation
When it is a positive integer, the exponent indicates how many copies of the base are multiplied together. For example, . The base appears times in the repeated multiplication, because the exponent is . Here, is the "base", is the "exponent", and is the "power" or, more specifically, "3 raised to the 5th power".
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.