oo is (1/ k )e-xfk. That is the density function of an exponential with mean k.
In some cases, as the parameters of a distribution go to infinity, the distribution converges to another distribution. C)a =e' a Equivalently, if cis a constant (not dependent on a), then ( r)a+c =e' lim 1+U-tOO a since we can set a'= a+ c, and r j(a'- c)--+ rfa' as a'--+ oo. As a simple example (not in the textbook) of a limiting distribution, consider a gamma distribution with a fixed mean fl• and let a--+ oo. Then 8 = Jlf a. t. which is the moment generating function of the constant fl. So as a --+ oo, the limiting distribution of a gamma is a distribution equal to the mean with probability 1.