Set partitions and moments of random variables |
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Authors: | Jesús de la Cal Javier Cárcamo |
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Institution: | a Departamento de Matemática Aplicada y Estadística e Investigación Operativa, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain b Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain |
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Abstract: | It is well known that the sequence of Bell numbers (Bn)n?0 (Bn being the number of partitions of the set n]) is the sequence of moments of a mean 1 Poisson random variable τ (a fact expressed in the Dobiński formula), and the shifted sequence (Bn+1)n?0 is the sequence of moments of 1+τ. In this paper, we generalize these results by showing that both and (where is the number of m-partitions of n], as they are defined in the paper) are moment sequences of certain random variables. Moreover, such sequences also are sequences of falling factorial moments of related random variables. Similar results are obtained when is replaced by the number of ordered m-partitions of n]. In all cases, the respective random variables are constructed from sequences of independent standard Poisson processes. |
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Keywords: | Set partitions Bell numbers Exponential generating function Moments of random variables Moment generating function Poisson process Poisson distribution |
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