The incomplete beta function, a generalization of the beta function, is defined as
For x = 1, the incomplete beta function coincides with the complete beta function. The relationship between the two functions is like that between the gamma function and its generalization the incomplete gamma function.
The regularized incomplete beta function (or regularized beta function for short) is defined in terms of the incomplete beta function and the complete beta function:
Working out the integral (one can use integration by parts) for integer values of a and b, one finds:
The regularized incomplete beta function is the cumulative distribution function of the Beta distribution, and is related to the cumulative distribution function of a random variable X from a binomial distribution, where the "probability of success" is p and the sample size is n:
Read more about this topic: Beta Function
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—Georges Bataille (1897–1962)
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