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:

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.

B_{1}(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

to

is called the **incomplete beta function ratio**. Represented by the symbol I_{x}, it is written as:

I_{x} (a, b) ≡ B_{x}(a, b) / B_{1}(a, b).

Where a > 0, b > 0 (DiDonato & Jarnagin, n.d.).

## Incomplete Beta Function Uses

The incomplete beta function and I_{x} 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

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.

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