Let A1,...,AN and B1,...,BM be two sequences of events and let νN(A) and νM(B) be the number of those Ai and Bj, respectively, that occur. Based on multivariate Lagrange interpolation, we give a method that yields linear bounds in terms of Sk,t, k+t ≤ m on the distribution of the vector (νN(A), νM(B)). For the same value of m, several inequalities can be generated and all of them are best bounds for some values of Sk,t. Known bivariate Bonferroni-type inequalities are reconstructed and new inequalities are generated, too.
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