Evaluation of Statistic Probability Distribution Functions: II. Chi-Squared Probability Function and the Inverse Functions of z, t, F, and Chi-Squared Distributions

Evaluation of probability distribution functions and their inverse functions plays a primarily important role in a statistical analysis and inference. In addition to statistically critical values, it produces probability values required to complete such inferential tasks. Therefore Chi-squared (χ²) distribution function was evaluated via its related incomplete gamma function and by integrating the power series and continued fraction expansion with the left-to-right and right-to-left approaches. The standard normal z, Student's t, F, and χ² values were computed using the step number-skipping and binary bisection search. With the optimized algorithm and a computer-based convergence technique as well as their simplicity, they resulted in a tremendous improvement of computational precision and efficiency. Other novel notion and important information on integrating probability distribution functions with computing tricks has also been introduced.