Several methods for the solution of the maximum likelihood equation, which arrises if parameters of the gamma and beta distribution are to be estimated, are presented and compared with respect to their applicability for computers. These procedures are based on interations using the NEWTON-RAPHSON method and the scoring method and the scoring method. On the other hand they can be based on a certain algorithm for interval bisection with subsequent inverse parabolic interpolation in order to determine the zeros of the maximum likelihood equation. A study concerning Monte Carlo simulation for the estimation methods and numerical tables are given.