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The combination of Mathematica's high algebraic capacity and method of generating functions is becoming an extremely efficient tool in probability theory and statistics. After an introductory example and a short overview on nonparametric methods, we show how generating functions of discrete statistics can be handled using Mathematica. Next, we solve two combinatorial problems, which are essential in calculating the generating function of nonparametric test statistics. Finally, these methods are used to produce accurate and extensive tables and plots of the distributions of some of the most popular nonparametric test statistics. Furthermore, we use graphs to illustrate approximations of the distributions of these test statistics by standard distributions.
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