# incomplete beta functions

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##### 2: 8.23 Statistical Applications
###### §8.23 Statistical Applications
The function $\mathrm{B}_{x}\left(a,b\right)$ and its normalization $I_{x}\left(a,b\right)$ play a similar role in statistics in connection with the beta distribution; see Johnson et al. (1995, pp. 210–275). …
##### 4: 8.1 Special Notation
Unless otherwise indicated, primes denote derivatives with respect to the argument. The functions treated in this chapter are the incomplete gamma functions $\gamma\left(a,z\right)$, $\Gamma\left(a,z\right)$, $\gamma^{*}\left(a,z\right)$, $P\left(a,z\right)$, and $Q\left(a,z\right)$; the incomplete beta functions $\mathrm{B}_{x}\left(a,b\right)$ and $I_{x}\left(a,b\right)$; the generalized exponential integral $E_{p}\left(z\right)$; the generalized sine and cosine integrals $\mathrm{si}\left(a,z\right)$, $\mathrm{ci}\left(a,z\right)$, $\mathrm{Si}\left(a,z\right)$, and $\mathrm{Ci}\left(a,z\right)$. Alternative notations include: Prym’s functions $P_{z}(a)=\gamma\left(a,z\right)$, $Q_{z}(a)=\Gamma\left(a,z\right)$, Nielsen (1906a, pp. 25–26), Batchelder (1967, p. 63); $(a,z)!=\gamma\left(a+1,z\right)$, $[a,z]!=\Gamma\left(a+1,z\right)$, Dingle (1973); $B(a,b,x)=\mathrm{B}_{x}\left(a,b\right)$, $I(a,b,x)=I_{x}\left(a,b\right)$, Magnus et al. (1966); $\mathrm{Si}\left(a,x\right)\to\mathrm{Si}\left(1-a,x\right)$, $\mathrm{Ci}\left(a,x\right)\to\mathrm{Ci}\left(1-a,x\right)$, Luke (1975).
##### 6: 8.18 Asymptotic Expansions of $I_{x}\left(a,b\right)$
8.18.1 $I_{x}\left(a,b\right)={\Gamma\left(a+b\right)x^{a}(1-x)^{b-1}}\*\left(\sum_{k=% 0}^{n-1}\frac{1}{\Gamma\left(a+k+1\right)\Gamma\left(b-k\right)}\left(\frac{x}% {1-x}\right)^{k}+O\left(\frac{1}{\Gamma\left(a+n+1\right)}\right)\right),$
##### 8: Bibliography P
• K. Pearson (Ed.) (1968) Tables of the Incomplete Beta-function. 2nd edition, Published for the Biometrika Trustees at the Cambridge University Press, Cambridge.
• H. N. Phien (1990) A note on the computation of the incomplete beta function. Adv. Eng. Software 12 (1), pp. 39–44.
• ##### 9: Bibliography N
• G. Nemes and A. B. Olde Daalhuis (2016) Uniform asymptotic expansion for the incomplete beta function. SIGMA Symmetry Integrability Geom. Methods Appl. 12, pp. 101, 5 pages.
• ##### 10: Bibliography D
• A. R. DiDonato and A. H. Morris (1992) Algorithm 708: Significant digit computation of the incomplete beta function ratios. ACM Trans. Math. Software 18 (3), pp. 360–373.
• J. Dutka (1981) The incomplete beta function—a historical profile. Arch. Hist. Exact Sci. 24 (1), pp. 11–29.