The computation of the Beta PDF is discussed and algorithms are described which permit calculation of the PDF over the full range of its parameters without recourse to approximations or polynomial expansions. This is achieved by giving simple rules for the treatment of the trivial cases and a new algorithm for the general cases, particularly for the cases where the PDF becomes infinity at the extremes of the range. The problems of underflow and overflow when performing the calculations on a digital computer are also addressed, and a general prescaling procedure is presented which overcomes these problems. It is shown that the new algorithms perform more efficiently than commercial mathematical packages.