In Part I the Maximum Likelihood/Entropy (ML/E) method of estimation of the cell probabilities for multinomial and contingency table problems is derived and discussed. This method is a generalization of the Maximum Likelihood estimator to situations when small probabilities are to be estimated and the standard Maximum Likelihood estimator is inadequate. In addition when no sample exists the technique gives meaningful results by reducing to the method of Maximum Entropy. The ML/E method is based on assuming an entropy prior on the cell probabilities and closely resembles the Pseudo-Bayes methods of Good, Fienberg, and Holland in which Dirichlet priors are assumed. Methods for calculating the ML/E estimates for varying circumstances including multidimensional tables are presented. Comparisons with other estimation methods are made and recommendations for selection of the more appropriate method in particular situations are given. In Part II we consider the Kolmogorov-Smirnov one-sample statistic. Various methods for calculating the Kolmogorov-Smirnov one-sample statistics have been developed in the literature. A transformation of an approximation method is here derived and some of its properties discussed. The main value of the new formulae is to obtain better convenient approximations in the lower tail than have been possible using other methods. The formulae are related to the theta functions. The relationships between various methods are given, as well as recommendations for each method of a usable range of the independent variable. An analysis is made-of the errors obtained by use of the approximation. / Ph. D.