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Beta Function
The Beta function (also called the Euler Integral of the first kind) is a definite integral, related to the gamma function. The most common representation for the function is:
The function goes by many different names. As is is usually defined by the above integral, it is sometimes called the “beta integral.” It’s also called the Euler Β-function, and is sometimes simply denoted by its variables: Β(p, q).
Other equivalent forms of the function exist, and are obtained by changing variables. You can find a list of various forms (including trigonometric forms) in the Digital Library of Mathematical Functions.
Practical Uses
The function is used in physics and string theory, where it can model properties of strong nuclear force. In time management problems, the beta distribution (which is the integrand of the Beta function) can be used to estimate average time to complete tasks. In calculus, the function is especially important because many other definite integrals can be rewritten as a Beta function: this useful general function can simplify the evaluation of many complicated integrals.
In probability and statistics, the Beta has many applications. It is sometimes used as a normalizing constant or as part of some definitions for probability distributions. For example, the probability mass function (PMF) for the Yule-Simon distribution uses the Β-function for its PMF. The function can also define a binomial coefficient, after adjusting indices. These statistical applications extend to a wide variety of real life applications, including modeling experimental frequency distributions of relative sunshine and humidity.
History
While Euler first developed the function, it was the French mathematician Jacques P.M. Binet who first used the beta symbol for the function.
Incomplete Beta Function
The incomplete beta function (also called the Euler Integral) is a generalized β-function; An independent integral (with integral bounds from 0 to x) replaces the definite integral. The formula is:
Where:
- 0 ≤ x ≤ 1,
- a, b > 0. Note: The definition is sometimes written to include negative integers (e.g. Özçag et al., 2008) but this isn’t commonplace.
B1(p, q) is the (complete) beta function; in other words, the function becomes complete as x = 1. The incomplete beta function can also be expressed in terms of the beta function or three complete gamma functions (DiDonato & Jarnagin, 1972).
Incomplete Beta-Function Ratio
The ratio of
to
is called the incomplete beta function ratio. Represented by the symbol Ix, it is written as:
Ix (a, b) ≡ Bx(a, b) / B1(a, b).
Where a > 0, b > 0 (DiDonato & Jarnagin, n.d.).
Incomplete Beta Function Uses
The incomplete beta function and Ix crop up in various scientific applications, including atomic physics, fluid dynamics, lattice theory (the study of lattices) and transmission theory (DiDonato & Morris, 1988):
- Calculating confidence intervals for t-tests, F-tests (Besset, 2001) and those based on the binomial distribution, where the incomplete beta function is used to calculate the limits (Young et al., 1998),
- Computing the probability in a binomial distribution tail (DTIC, 1979),
- Creating cumulative probabilities for the standard normal distribution (Klugman, 2013).
- Finding a measurement larger than a certain value, for data following a beta distribution.
References
Alhassan, E. et al. (2015). On some Applications of Beta Function in some Statistical Distributions. Researcher.
Besset, D. (2001). Object-Oriented Implementation of Numerical Methods. An Introduction with Java & Smalltalk. Elsevier Science.
DiDonato, A. & Jarnagin, M. (n.d.). The Efficient Calculation of the Incomplete
Beta-Function Ratio for Half-Integer Values of the Parameters a, b. Retrieved September 21, 2020 from: https://www.ams.org/journals/mcom/1967-21-100/S0025-5718-1967-0221730-X/S0025-5718-1967-0221730-X.pdf
DiDonato, A. & Jarnagin, M. (1972). A Method for Computing the Incomplete Beta Function Ratio. U.S. Naval Weapons Laboratory. Retrieved September 21, 2020 from: https://apps.dtic.mil/dtic/tr/fulltext/u2/642495.pdf
DiDonato, A. & Morris, F. (1988). Significant Digit Computation of the IBF. Retrieved September 21, 2020 from: https://apps.dtic.mil/dtic/tr/fulltext/u2/a210118.pdf (PDF).
DTIC–Defense Technical Information Center (1979). A Note on the Incomplete Beta Function.
Klugman, S. et al. (2013). Loss Models. Wiley.
Özçag, E et al. (2008). An extension of the incomplete beta function for negative integers, J. Math. Anal. Appl.
Young, L. et al. (1998). Statistical Ecology. Springer.
Riddhi, D. Beta Function and its Applications. Retrieved December 5, 2019 from: http://sces.phys.utk.edu/~moreo/mm08/Riddi.pdf
Factorial, Gamma and Beta Functions. Retrieved December 6, 2019 from: http://www.mhtlab.uwaterloo.ca/courses/me755/web_chap1.pdf