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Abstract
The principal objective of this study has been to derive and develop algorithms to approximate probabilities, and when appropriate, their inverse elements, in the Gaussian, Chi-square, and Student's t distributions. These algorithms have been specifically designed for use by the newly-developed electronic hand calculator, although they are equally applicable to the more sophisticated computer.
The disclosure of these algorithms is timely because it coincides with the appearance of the widely-accepted hand calculator. One might describe the relationship between algorithm and calculator as symbiotic, in the sense that each needs the other. The calculator needs the input of the algorithm; the algorithm cannot function without the calculator.
A secondary objective of this study was the directed search of the literature to uncover, if possible, published formulas capable of approximating probabilities and their inverse in the Gaussian, Chi-square, and Student's t distributions. If such a search would reveal the existence of competing formulas capable of successfully rivaling the algorithms presented in this study in simplicity of structure, ease and rapidity of performance, range of applicability, and accuracy, there would be no justification for this study nor for the algorithms arising from this study.
The literature did indeed disclose the existence of published formulas capable of fulfilling one or more, but not all four of the criteria, in the Gaussian distribution. However, none of the rival formulas could challenge the algorithms submitted in this treatise, in performance and versatility, in meeting all four of the criteria in all three of the distributions. The ineptitude of the rival formulas is most clearly manifested in their functional limitations in the Chi-square and Student's t distributions.
The objectives of this study have been consummated. Structures and the strategies used in development of these structures are described, as well as the description of the application of these algorithms to typical examples, defining the range of their usefulness, and assessing their accuracy.
Of the twenty-three algorithms presented in this study, five deal with the Gaussian distribution, eight with the Chi-square distribution, and ten with Student's t test.
