A Cumulative Distribution Function (CDF) is the integral of its respective probability distribution function (PDF). CDFs are usually well behaved functions with values in the range [0,1]. CDFs are important in computing critical values, P-values and power of statistical tests.

Many CDFs are computed directly from closed form expressions. Others can be difficult to compute because they involve evaluating a very large number states, e.g., the CDFs of the Friedman or USquared distributions. In these cases you have the following options:

• Use a built-in table that consists of a limited number of exact (precomputed) values.
• Compute an approximate CDF based on the prevailing approximation method
• Use a Monte-Carlo approach.
• Attempt to compute an exact CDF.

Built-in tables are ideal if they cover the range of the parameters that you need. Monte-Carlo methods can be tricky in the sense that repeated application may return small variations in values. Computing the exact CDF may be nice but could sometimes be impractical. In most situations the range of parameters that is practical to compute on a desktop machine is already covered in the built-in tables. Larger parameters have not been considered because they may take a long time to compute or because they may require 64 bit processors. In addition, most approximations tend to improve with increasing size of the parameters.