The incomplete beta function (also called the Euler Integral) is a generalized beta function; An independent integral (with integral bounds from 0 to x) replaces the definite integral. The formula is:
- 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 gamma 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
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.
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.
Stephanie Glen. "Incomplete Beta Function / Integral" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/incomplete-beta-function-integral/
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