This function evaluates a general continuous cumulative distribution function given ordinates of the density.

Function GCDF evaluates a continuous distribution function, given ordinates of the probability density function. It requires that the range of the distribution be specified in X. For distributions with infinite ranges, endpoints must be chosen so that most of the probability content is included. The function GCDF first fits a curve to the points given in X and F with either a piecewise linear interpolant or a C 1 cubic spline interpolant based on a method by Akima (1970). Function GCDF then determines the area, A, under the curve. (If the distribution were of finite range and if the fit were exact, this area would be 1.0.) Using the same fitted curve, GCDF next determines the area up to the point x0 (= X0). The value returned is the area up to x0 divided by A. Because of the scaling by A, it is not assumed that the integral of the density defined by X and F is 1.0. For most distributions, it is likely that better approximations to the distribution function are obtained when IOPT equals 3 or 4, that is, when a cubic spline is used to approximate the function. It is also likely that better approximations can be obtained when the abscissas are chosen more densely over regions where the density and its derivatives (when they exist) are varying greatly.

In this example, we evaluate the beta distribution function at the point 0.6. The probability density function of a beta random variable with parameters p and q is

where Γ(⋅) is the gamma function. The density is equal to 0 outside the interval [0, 1]. We compute a constant multiple (we can ignore the constant gamma functions) of the density at 300 equally spaced points and input this information in X and F. Knowing that the probability density of this distribution is very peaked in the vicinity of 0.5, we could perhaps get a better fit by using unequally spaced abscissas, but we will keep it simple. Note that this is the same example as one used in the description of routine BETDF. The result from BETDF would be expected to be more accurate than that from GCDF since BETDF is designed specifically for this distribution.