The regularized incomplete beta function (also called the regularized beta function) is widely used in statistics and some areas of physics as the cumulative distribution function of the beta distribution . It represent the CDF for a random number Y that obeys the beta distribution .
The regularized incomplete beta function is used when it’s inconvenient to calculate the incomplete beta function on its own. The “regularized” version divides this function by the complete beta function , hence another alternate name — the incomplete beta function ratio .
Formula for the Regularized Incomplete Beta Function
The regularized incomplete beta function is defined in terms of the incomplete beta function B(z; a, b) — also called the Euler integral — and the complete beta function b (a. b) :
The basic definition shown above cannot be used for nonpositive integers a or b, as this will lead to an indeterminate expression. For nonpositive integers, you may need to use a more complete definition such as :
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