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An inductive derivation of Stirling numbers of the second kind and their applications in statistics. (English) Zbl 0910.62015

We present an inductive proof to derive Stirling numbers of the second kind. We also demonstrate the application of Stirling numbers in calculating moments of some discrete distributions. The usual method of deriving raw moments of higher order of an integer-valued random variable is to derive the moment generating function and then differentiate as many times as the order of the moment required. We take an advantage of factorial moments which are easily derived for integer-valued random variables. These moments can be combined with the help of Stirling numbers of the second kind for deriving raw moments and hence central moments of integer-valued random variables. The advantage of the method presented here for calculated raw moments or central moments of a distribution is that it avoids using derivatives of higher order.

MSC:

62E15 Exact distribution theory in statistics
11B73 Bell and Stirling numbers
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