A Probabilistic Derivation of Stirling's Formula

Stirlings formula is one of the most frequently used results from asymptotics. It is used in probability and statistics, algorithm analysis and physics. In this thesis we shall give a new probabilistic derivation of Stirlings formula. Our motivation comes from sampling randomly with replacement from a group of n distinct alternatives. Usually a repetition will occur before we obtain all n distinct alternatives consecutively. We shall show that Stirlings formula can be derived and interpreted as follows as n---infinity the expected total number of distinct alternatives we must sample before all n are obtained consecutively is asymptotically equal to the expected number of attempts we make to obtain all n distinct alternatives consecutively times the expected number of distinct alternatives obtained per attempt.