Cumulative distribution functions and moments of weighted lattice polynomials

In the first part of this presentation we define the concept of weighted lattice polynomials as lattice polynomials constructed from both variables and parameters. We then show that, in any bounded distributive lattice, these functions can always be written in conjunctive and disjunctive normal forms. We also show that these functions include the class of discrete Sugeno integrals and that they are characterized by a remarkable median based decomposition formula.
In the second part we give the cumulative distribution functions, the expected values, and the moments of weighted lattice polynomials when regarded as real functions. Since weighted lattice polynomial functions include Sugeno integrals, lattice polynomial functions, and order statistics, our results encompass the corresponding formulas for these particular functions. We then conclude with some applications of our results to the reliability analysis of coherent systems.