Computation of the Chi-Square and Poisson Distribution

The paper deals with the computation of upper and lower tail probabilities of the chi-square and Poisson distribution with a specified relative accuracy on both tails for virtually all possible parameter values. With some supplement the proposed algorithms will also work for the general gamma distribution. If the parameters are small, open forward and backward recursion is used for the summation with an adaptive number of steps depending on the specified accuracy. For large parameters asymptotic expansions related to the central limit theorem are applied for the approximation. The basic ideas of the proposed methods will also be applicable to other elementary statistical distributions such as the binomial, beta, and F-distribution as well as the hypergeometric distribution.