The incomplete beta function and its ratio to the complete beta function

Authors:
David Osborn and Richard Madey

Journal:
Math. Comp. **22** (1968), 159-162

DOI:
https://doi.org/10.1090/S0025-5718-68-99888-8

Full-text PDF Free Access

Abstract | References | Additional Information

Abstract: The incomplete beta function, , and its ratio to the complete beta function has been calculated on an IBM-7044 computer to five significant figures and tabulated for the arguments *p* and *q* each in the interval from 0.5 to 2.0 with increments of 0.05 and for the parameter in the interval 0.1 to 1.0 with increments of 0.01. The function was evaluated by first expanding a factor of the integrand in a binomial series and then integrating each term. In order to facilitate computation, one expansion is taken for and another for . The expansions are truncated when the relative error is less than or equal to an arbitrarily predetermined fraction. Typical running time for achieving five significant figures on the IBM-7044 is about one minute for values.

**[1]**Richard Madey & T. E. Stephenson, "Quality factors for degraded proton spectra,"*Proceedings of the Second Symposium on Protection Against Radiations in Space*(held at Gatlinburg, Tennessee, October 1964), NASA SP-71, 229, 1965.**[2]**E. T. Whittaker & G. M. Watson,*A Course of Modern Analysis*, 4th ed., Cambridge Univ. Press, New York, 1927; 1946, pp. 253-254.**[3]***Handbook of mathematical functions, with formulas, graphs and mathematical tables*, Edited by Milton Abramowitz and Irene A. Stegun. Fifth printing, with corrections. National Bureau of Standards Applied Mathematics Series, Vol. 55, National Bureau of Standards, Washington, D.C., (for sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 20402), 1966. MR**0208798****[4]***Tables of the incomplete beta-function*, Originally prepared under the direction of and edited by Karl Pearson. Second edition with a new introduction by E. S. Pearson and N. L. Johnson, Published for the Biometrika Trustees at the Cambridge University Press, London, 1968. MR**0226815****[5]***Tables of the Binomial Probability Distribution*, National Bureau of Standards, Applied Mathematics Series, No. 6, United States Government Printing Office, Washington, D. C., 1950. MR**0035108****[6]**L. E. Simon & F. E. Grubbs,*Tables of the Cumulative Binomial Probabilities*, Ballistic Research Laboratories Ordinance Corps. Pamphlet ORDP20-1, Aberdeen Proving Ground, Maryland, 1952 and 1956 Supplement.**[7]**Harry G. Romig,*50–100 binomial tables*, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1953. MR**0056358****[8]***Tables of the cumulative binomial probability distribution*, The Annals of the Computation Laboratory of Harvard University, vol. 35, Harvard University Press, Cambridge, Mass., 1955. MR**0082203****[9]**Sol Weintraub,*Tables of the cumulative binomial probability distribution for small values of 𝑝*, The Free Press of Glencoe, New York; Collier-Macmillan, Ltd., London, 1963. MR**0158449****[10]**Ross R. Middlemiss,*Differential and Integral Calculus*, McGraw-Hill Book Company, Inc., New York, 1940. MR**0001793****[11]**R. S. Burington & D. C. May,*Handbook of Probability and Statistics with Tables*, Handbook Publishers, Sandusky, Ohio, 1953.**[12]**A. Erdélyi, W. Magnus, F. Oberhettinger & F. Tricomi, Higher Transcendental Functions, Vol. 1, Bateman Manuscript Project, McGraw-Hill, New York, 1953. MR**15**, 419.**[13]**A. Fletcher, J. C. P. Miller, and L. Rosenhead,*An Index of Mathematical Tables*, McGraw-Hill Book Company, New York; Scientific Computing Service Limited, London, 1946. MR**0018419****[14]**J. Arthur Greenwood and H. O. Hartley,*Guide to tables in mathematical statistics*, Princeton University Press, Princeton, N.J., 1962. MR**0154350****[15]**Karl Pearson,*Tables for Statisticians and Biometricians*, Cambridge Univ. Press, New York, 1914.**[16]**Karl Pearson,*Tables for Statisticians and Biometricians*, Part II, London, 1935, p. 240.**[17]**Karl Pearson & Margaret Pearson, "On the numerical evaluation of high order incomplete Eulerian integrals,"*Biometrika*, v. 27, 1935, pp. 409-423.**[18]***Reference Handbook of the Boulder Laboratories Library*. Part III:*Mathematical Tables*, 2nd ed., U. S. Dept. of Commerce, National Bureau of Standards, Boulder, Colorado, 1964.**[19]**H. E. Soper,*The Numerical Evaluation of the Incomplete B-Function*, Cambridge Tracts for Computers, No. 7, Cambridge Univ. Press, New York, 1921.**[20]**J. Wishart, "On the approximate quadrature of certain skew curves, with an account of the researches of Thomas Bayes,"*Biometrika*, v. 19, 1927, pp. 1-38.**[21]**Richard Madey & David Osborn,*Tables of the Incomplete Beta Function and Its Ratio to the Complete Beta Function*. (Unpublished work copyrighted 1967.)

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-68-99888-8

Article copyright:
© Copyright 1968
American Mathematical Society