The beta function is also closely related to binomial coefficients. When m (or n, by symmetry) is a positive integer, it follows from the definition of the gamma function Γ that[1]
Relationship to the gamma function
A simple derivation of the relation can be found in Emil Artin's book The Gamma Function, page 18–19.[2]
To derive this relation, write the product of two factorials as integrals. Since they are integrals in two separate variables, we can combine then into an iterated integral:
Changing variables by u = st and v = s(1 − t), because u + v = s and u / (u+v) = t, we have that the limits of integrations for s are 0 to ∞ and the limits of integration for t are 0 to 1. Thus produces
Dividing both sides by gives the desired result.
The stated identity may be seen as a particular case of the identity for the integral of a convolution. Taking
The beta function satisfies several identities analogous to corresponding identities for binomial coefficients, including a version of Pascal's identity
The incomplete beta function, a generalization of the beta function, is defined as[7][8]
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. For positive integer a and b, the incomplete beta function will be a polynomial of degree a + b - 1 with rational coefficients.
By the substitution and , we show that
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:
converges rapidly when is not close to 1. The and convergents are less than , while the and convergents are greater than .
For , the function may be evaluated more efficiently using .[8]
Multivariate beta function
The beta function can be extended to a function with more than two arguments:
This multivariate beta function is used in the definition of the Dirichlet distribution. Its relationship to the beta function is analogous to the relationship between multinomial coefficients and binomial coefficients. For example, it satisfies a similar version of Pascal's identity:
Even if unavailable directly, the complete and incomplete beta function values can be calculated using functions commonly included in spreadsheet or computer algebra systems.
In Microsoft Excel, for example, the complete beta function can be computed with the GammaLn function (or special.gammaln in Python'sSciPy package):
Value = Exp(GammaLn(a) + GammaLn(b) − GammaLn(a + b))
This result follows from the properties listed above.
The incomplete beta function cannot be directly computed using such relations and other methods must be used. In GNU Octave, it is computed using a continued fraction expansion.
The incomplete beta function has existing implementation in common languages. For instance, betainc (incomplete beta function) in MATLAB and GNU Octave, pbeta (probability of beta distribution) in R, or special.betainc in SciPy compute the regularized incomplete beta function—which is, in fact, the cumulative beta distribution—and so, to get the actual incomplete beta function, one must multiply the result of betainc by the result returned by the corresponding beta function. In Mathematica, Beta[x, a, b] and BetaRegularized[x, a, b] give and , respectively.
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