226 Chapter 6.Special Functions TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT (0.5,5.0) (8.010.0) /(1.0,3.0) Permission is 6 (0.5,0.5) .com or call 1-800-872- granted for (including this one) internet to any server computer, t users to make one paper 1988-1992 by Cambridge University Press. from NUMERICAL RECIPES IN C: (5.0,0.5) 0 (North America THE 0 .6 Programs Figure 6.4.1.The incomplete beta function (,b)for five different pairs of (a,b).Notice that the pairs (0.5,5.0)and(5.0,0.5)are symmetrically related as indicated in equation (6.4.3). 6.4 Incomplete Beta Function,Student's Distribution,F-Distribution,Cumulative 1788-1982 ART OF SCIENTIFIC COMPUTING (ISBN 0-521 Binomial Distribution The incomplete beta function is defined by Numerical Recipes -43108-5 I(a,b)= Ba,0=1 B(a,b) B(a,b)Jo ta-1(1-t)b-1dt (a,b>0) (6.4.1) (outside Software. It has the limiting values North Io(a,b)=0 11(a,b)=1 (6.4.2) visit website f machine and the symmetry relation Iz(a,b)=1-11-x(b,a) (6.4.3) If a and b are both rather greater than one,then I(a,b)rises from"near-zero"to "near-unity"quite sharply at about x =a/(a+b).Figure 6.4.1 plots the function for several pairs (a,b).226 Chapter 6. Special Functions Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machine￾readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). 0 (5.0,0.5) (0.5,0.5) (8.0,10.0) (1.0,3.0) (0.5,5.0) .2 .4 .6 1 .8 0 .2 .4 .6 .8 1 incomplete beta function Ix(a,b) x Figure 6.4.1. The incomplete beta function Ix(a, b) for five different pairs of (a, b). Notice that the pairs (0.5, 5.0) and (5.0, 0.5) are symmetrically related as indicated in equation (6.4.3). 6.4 Incomplete Beta Function, Student’s Distribution, F-Distribution, Cumulative Binomial Distribution The incomplete beta function is defined by Ix(a, b) ≡ Bx(a, b) B(a, b) ≡ 1 B(a, b)
x 0 t a−1(1 − t) b−1dt (a, b > 0) (6.4.1) It has the limiting values I0(a, b)=0 I1(a, b)=1 (6.4.2) and the symmetry relation Ix(a, b)=1 − I1−x(b, a) (6.4.3) If a and b are both rather greater than one, then I x(a, b) rises from “near-zero” to “near-unity” quite sharply at about x = a/(a + b). Figure 6.4.1 plots the function for several pairs (a, b)