Numerical recipes

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 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 ν 2
ν 2 +ν 1 F
� �
ν2 ν1
,
2 2
Cumulative Binomial Probability Distribution
(6.4.11)
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 x(a, b) as follows:
P ≡
n�
j=k
� �
n
p
j
j (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 Mathematics
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: Cambridge
University Press).
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