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
Famous quotes containing the words incomplete and/or function:
“Each of us is incomplete compared to someone else, an animal’s incomplete compared to a person ... and a person compared to God, who is complete only to be imaginary.”
—Georges Bataille (1897–1962)
“Morality and its victim, the mother—what a terrible picture! Is there indeed anything more terrible, more criminal, than our glorified sacred function of motherhood?”
—Emma Goldman (1869–1940)