Exponential moments from the transform

Topic: Random variables and distributions · more questions on this topic
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Let X be exponential with rate 1. Use the moment generating function to show that \mathbb{E} X^n=n!.

Worked answer (click to show)

The moment generating function of the rate-1 exponential law is M(s)=1/(1-s) for s<1. Differentiating n times gives M^{(n)}(s)=n!/(1-s)^{n+1}. The moment formula M^{(n)}(0)=\mathbb{E} X^n therefore yields n!.