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Evaluation of Statistic Probability Distribution Functions: I. Computation of the Standard Normal z, Student's t, F, and Chi-Squared Values

Evaluation of probability distribution functions and their inverse functions play a central role in finding a critical value or threshold for a statistical inference and analysis. A polynomial fitting of the required number of recurrences for evaluating standard normal function with a continued fraction expansion was created for a desirable precision of computational results. Student's t and the F distribution functions were evaluated by their related incomplete beta function and by combining the left-to-right approach with the right-to-left one to a continued fraction expansion. They result in a tremendous improvement of evaluation precision and efficiency with the optimized algorithm and a computing convergence technique as well as their simplicity. Other novel notion and important information on integrating probability distribution functions with computing tricks have also been provided.