If x is a r. v. such that e(x^n)=n! find the m. g.f. of x, mx(t). also find the ch. f. of x, and from this deduce the distribution of x.

provided that .

Similarly,







You can find the CDF/PDF using any of the various inversion formulas. One way would be to compute



The integral can be rewritten as



so that



There are lots of ways to compute this integral. For instance, you can take the Laplace transform with respect to , which gives




and taking the inverse transform returns



which describes an exponential distribution with parameter .