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Beta function
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==Incomplete beta function== The '''incomplete beta function''', a generalization of the beta function, is defined as<ref>{{citation | last1 = Zelen | first1 = M. | last2 = Severo | first2 = N. C. | editor1-last = Abramowitz | editor1-first = Milton | editor1-link = Milton Abramowitz | editor2-last = Stegun | editor2-first = Irene A. | editor2-link = Irene Stegun | year = 1972 | title = [[Abramowitz and Stegun|Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables]] | chapter = 26. Probability functions | pages = [https://archive.org/details/handbookofmathe000abra/page/944 944] | publisher = [[Dover Publications]] | location = New York | isbn = 978-0-486-61272-0}}</ref><ref name="paris-ibf">{{dlmf|mode=cs2 | last = Paris | first = R. B. | id = 8.17 | title = Incomplete beta functions}}</ref> :<math> \Beta(x;\,a,b) = \int_0^x t^{a-1}\,(1-t)^{b-1}\,dt. </math> For {{math|''x'' {{=}} 1}}, the incomplete beta function coincides with the complete beta function. For positive integers ''a'' and ''b'', the incomplete beta function will be a polynomial of degree ''a'' + ''b'' - 1 with rational coefficients. By the substitution <math>t = \sin^2\theta</math> and <math>t = \frac1{1+s}</math>, we can show that :<math>\Beta(x;\,a,b) = 2 \int_0^{\arcsin \sqrt x} \sin^{2a-1\!}\theta\cos^{2b-1\!}\theta\,\mathrm d\theta = \int_{\frac{1-x}x}^\infty \frac{s^{b-1}}{(1+s)^{a+b}}\,\mathrm ds</math> The '''regularized incomplete beta function''' (or '''regularized beta function''' for short) is defined in terms of the incomplete beta function and the complete beta function: :<math> I_x(a,b) = \frac{\Beta(x;\,a,b)}{\Beta(a,b)}. </math> The regularized incomplete beta function is the [[cumulative distribution function]] of the [[beta distribution]], and is related to the [[cumulative distribution function]] <math>F(k;\,n,p)</math> of a [[random variable]] {{mvar|X}} following a [[binomial distribution]] with probability of single success {{mvar|p}} and number of Bernoulli trials {{mvar|n}}: :<math>F(k;\,n,p) = \Pr\left(X \le k\right) = I_{1-p}(n-k, k+1) = 1 - I_p(k+1,n-k). </math> ===Properties=== <!-- (Many other properties could be listed here.)--> :<math>\begin{align} I_0(a,b) &= 0 \\ I_1(a,b) &= 1 \\ I_x(a,1) &= x^a\\ I_x(1,b) &= 1 - (1-x)^b \\ I_x(a,b) &= 1 - I_{1-x}(b,a) \\ I_x(a+1,b) &= I_x(a,b)-\frac{x^a(1-x)^b}{a \Beta(a,b)} \\ I_x(a,b+1) &= I_x(a,b)+\frac{x^a(1-x)^b}{b \Beta(a,b)} \\ \int \Beta(x;a,b) \mathrm{d}x &= x \Beta(x; a, b) - \Beta(x; a+1, b) \\ \Beta(x;a,b)&=(-1)^{a} \Beta\left(\frac{x}{x-1};a,1-a-b\right) \end{align}</math> ===Continued fraction expansion=== The [[generalized continued fraction|continued fraction]] expansion :<math>\Beta(x;\,a,b) = \frac{x^{a} (1 - x)^{b}}{a \left( 1 + \frac{{d}_{1}}{1 +} \frac{{d}_{2}}{1 +} \frac{{d}_{3}}{1 +} \frac{{d}_{4}}{1 +} \cdots \right)}</math> with odd and even coefficients respectively :<math>{d}_{2 m + 1} = - \frac{(a + m) (a + b + m) x}{(a + 2 m) (a + 2 m + 1)}</math> :<math>{d}_{2 m} = \frac{m (b - m) x}{(a + 2 m - 1) (a + 2 m)}</math> converges rapidly when <math>x</math> is not close to 1. The <math>4 m</math> and <math>4 m + 1</math> convergents are less than <math>\Beta(x;\,a,b)</math>, while the <math>4 m + 2</math> and <math>4 m + 3</math> convergents are greater than <math>\Beta(x;\,a,b)</math>. For <math>x > \frac{a + 1}{a + b + 2}</math>, the function may be evaluated more efficiently using <math>\Beta(x;\,a,b) = \Beta(a, b) - \Beta(1 - x;\,b,a)</math>.<ref name="paris-ibf"/>
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