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In many applications, like database management systems, is very useful to have an expression to compute the cardinality of cartesian product of k sets without repeated elements; we designate this problem as T (k). The value of | T (k) | is upper-bounded by the multiplication of cardinalities of the sets. As long as we have searched, it has not been reported a general expression to compute T (k) using cardinalities of the intersections of sets, this is the main topic of this paper. Given three sets with indices { 0 , 1 , 2 } , C i is the cardinality of one set, C i , j (i < j) and C i , j , l (i < j < l) are respectively the cardinalities of the intersections of 2 and 3 sets, then the searched formulas for T (k) are: T (1) = C 0 ; T (2) = C 0 C 1 - C 0 , 1 ; T (3) = C 0 C 1 C 2 - (C 0 , 1 C 2 + C 0 , 2 C 1 + C 1 , 2 C 0) + 2 C 0 , 1 , 2 . In this paper, we prove formulas for computing T (k) and its specialization when a set is contained in the next sets. For this purpose, we will use concepts like partitions of the integer k in v parts, Bell numbers, Stirling numbers of the first kind and Stirling numbers of the second kind. Additionally, we present a complexity analysis for the computation of T (k). [ABSTRACT FROM AUTHOR]

In this paper, two new maximum likelihood methods are proposed to estimate the unknown sample size n in a simple random sample from an absolutely continuous population. These methods are based on the number of records and the record values in the sample. The use of these methods is explored through simulations. An application to a real data set is also included. [ABSTRACT FROM AUTHOR]

For a wide class of discrete distributions, we derive a representation of the inverse (negative) moments through the Stirling numbers of the first kind and inverse factorial moments. We specialize the results for the Poisson, binomial, hypergeometric and negative binomial distributions. [ABSTRACT FROM AUTHOR]

Through the use of the Stirling numbers of the first kind, a general class of approximations is derived for the negative moments of the Poisson distribution. [Copyright &y& Elsevier]