6.4 Incomplete Beta Function 229 So,you can use(6.4.9)and the above routine betai to evaluate the function. F-Distribution Probability Function This function occurs in the statistical test of whether two observed samples have the same variance.A certain statistic F,essentially the ratio of the observed dispersion of the first sample to that of the second one,is calculated.(For further details,see Chapter 14.)The probability that F would be as large as it is if the first sample's underlying distribution actually has smaller variance than the second's is denoted (Fv1,v2),where vi and v2 are the number of degrees of freedom in the first and second samples,respectively.In other words,Q(Fv,v2)is the significance 虽 level at which the hypothesis"I has smaller variance than 2"can be rejected.A small numerical value implies a very significant rejection,in turn implying high confidence in the hypothesis"1 has variance greater or equal to 2." Q(Fv,v2)has the limiting values 袋 Q0,2)=1 Q(∞h,2)=0 (6.4.10) 2 Its relation to the incomplete beta function I(a.b)as evaluated by betai above is 、。 Press. Q(FlV1,v2)=I V2 V1 2+1F 2 (6.4.11) Programs SCIENTIFIC 6 Cumulative Binomial Probability Distribution Suppose an event occurs with probability p per trial.Then the probability P of its occurring k or more times in n trials is termed a cumulative binomial probability, and is related to the incomplete beta function I(a,b)as follows: 10.621 (1-p)n-j=I2(k,n-k+1) 43106 (6.4.12) E喜 Numerical Recipes (outside For n larger than a dozen or so,betai is a much better way to evaluate the sum in North Software. (6.4.12)than would be the straightforward sum with concurrent computation of the binomial coefficients.(For n smaller than a dozen,either method is acceptable. CITED REFERENCES AND FURTHER READING: Abramowitz,M.,and Stegun,I.A.1964.Handbook of Mathematical Functions,Applied Mathe- matics Series,Volume 55 (Washington:National Bureau of Standards;reprinted 1968 by Dover Publications,New York).Chapters 6 and 26. Pearson,E.,and Johnson,N.1968,Tables of the Incomplete Beta Function (Cambridge:Cam- bridge University Press).6.4 Incomplete Beta Function 229 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). So, you can use (6.4.9) and the above routine betai to evaluate the function. F-Distribution Probability Function This function occurs in the statistical test of whether two observed samples have the same variance. A certain statistic F, essentially the ratio of the observed dispersion of the first sample to that of the second one, is calculated. (For further details, see Chapter 14.) The probability that F would be as large as it is if the first sample’s underlying distribution actually has smaller variance than the second’s is denoted Q(F|ν1, ν2), where ν1 and ν2 are the number of degrees of freedom in the first and second samples, respectively. In other words, Q(F|ν 1, ν2)is the significance level at which the hypothesis “1 has smaller variance than 2” can be rejected. A small numerical value implies a very significant rejection, in turn implying high confidence in the hypothesis “1 has variance greater or equal to 2.” Q(F|ν1, ν2) has the limiting values Q(0|ν1, ν2)=1 Q(∞|ν1, ν2)=0 (6.4.10) Its relation to the incomplete beta function Ix(a, b) as evaluated by betai above is Q(F|ν1, ν2) = I ν2 ν2+ν1F ν2 2 , ν1 2 (6.4.11) Cumulative Binomial Probability Distribution Suppose an event occurs with probability p per trial. Then the probability P of its occurring k or more times in n trials is termed a cumulative binomial probability, and is related to the incomplete beta function Ix(a, b) as follows: P ≡ n j=k n j pj (1 − p) n−j = Ip(k, n − k + 1) (6.4.12) For n larger than a dozen or so, betai is a much better way to evaluate the sum in (6.4.12) than would be the straightforward sum with concurrent computation of the binomial coefficients. (For n smaller than a dozen, either method is acceptable.) CITED REFERENCES AND FURTHER READING: Abramowitz, M., and Stegun, I.A. 1964, Handbook of Mathematical Functions, Applied Mathe￾matics Series, Volume 55 (Washington: National Bureau of Standards; reprinted 1968 by Dover Publications, New York), Chapters 6 and 26. Pearson, E., and Johnson, N. 1968, Tables of the Incomplete Beta Function (Cambridge: Cam￾bridge University Press).