Appendix B
The beta function is defined by
β ( a , b ) = ∫ 0 1  t a − 1   ( 1 − t )  b − 1   d t       ( 70 )      MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFYoGycqGGOaakcqWGHbqycqGGSaalcqWGIbGycqGGPaqkcqGH9aqpdaWdXaqaaiabdsha0naaCaaaleqabaGaemyyaeMaeyOeI0IaeGymaedaaOGaeiikaGIaeGymaeJaeyOeI0IaemiDaqNaeiykaKYaaWbaaSqabeaacqWGIbGycqGHsislcqaIXaqmaaGccqWGKbazcqWG0baDcaWLjaGaaCzcaiabcIcaOiabiEda3iabicdaWiabcMcaPaWcbaGaeGimaadabaGaeGymaedaniabgUIiYdaaaa@4D5E@
for all a, b > 0. The incomplete beta function for all x ∈ [0,1] is then defined by
β ( x , a , b ) = ∫ 0 x  t a − 1      ( 1 − t )  b − 1   d t       ( 71 )   MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFYoGycqGGOaakcqWG4baEcqGGSaalcqWGHbqycqGGSaalcqWGIbGycqGGPaqkcqGH9aqpdaWdXaqaaiabdsha0naaCaaaleqabaGaemyyaeMaeyOeI0IaeGymaedaaaqaaiabicdaWaqaaiabdIha4bqdcqGHRiI8aOGaeiikaGIaeGymaeJaeyOeI0IaemiDaqNaeiykaKYaaWbaaSqabeaacqWGIbGycqGHsislcqaIXaqmaaGccqWGKbazcqWG0baDcaWLjaGaaCzcaiabcIcaOiabiEda3iabigdaXiabcMcaPaaa@5037@
and
β −  ( x , a , b ) = β ( a , b ) − β ( x , a , b )         ( 72  )     = ∫ x 1  t a − 1      ( 1 − t )  b − 1   d t   ( 73  )        MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeGabaabfuaabmqaciaaaeaaiiGacqWFYoGydaahaaWcbeqaaiabgkHiTaaakiabcIcaOiabdIha4jabcYcaSiabdggaHjabcYcaSiabdkgaIjabcMcaPiabg2da9iab=j7aIjabcIcaOiabdggaHjabcYcaSiabdkgaIjabcMcaPiabgkHiTiab=j7aIjabcIcaOiabdIha4jabcYcaSiabdggaHjabcYcaSiabdkgaIjabcMcaPiaaxMaacaWLjaaabaWaaeWaaeaacqaI3aWncqaIYaGmaiaawIcacaGLPaaaaeaacqGH9aqpdaWdXaqaaiabdsha0naaCaaaleqabaGaemyyaeMaeyOeI0IaeGymaedaaaqaaiabdIha4bqaaiabigdaXaqdcqGHRiI8aOGaeiikaGIaeGymaeJaeyOeI0IaemiDaqNaeiykaKYaaWbaaSqabeaacqWGIbGycqGHsislcqaIXaqmaaGccqWGKbazcqWG0baDaeaadaqadaqaaiabiEda3iabiodaZaGaayjkaiaawMcaaaaaaaa@66A5@
Using a continued fraction representation, these functions can be quickly numerically evaluated in O(max⁡(a,b) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaGcaaqaaiGbc2gaTjabcggaHjabcIha4jabcIcaOiabdggaHjabcYcaSiabdkgaIjabcMcaPaWcbeaaaaa@3617@) in the worst case [15, Chapter 6].
A great interest of this function is that it is connected to the cumulative distribution function of a binomial distribution by the following relation:
ℙ ( ℬ ( n , p ) ≥ k ) = β ( p , k , n − k + 1 )  β ( k , n − k + 1 )         ( 74 )   MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaWaaeGaeaaakeaatuuDJXwAK1uy0HMmaeXbfv3ySLgzG0uy0HgiuD3BaGqbaiab=LriqjabcIcaOGWaaiab+XsicjabcIcaOiabd6gaUjabcYcaSiabdchaWjabcMcaPiabgwMiZkabdUgaRjabcMcaPiabg2da9maalaaabaacciGae0NSdiMaeiikaGIaemiCaaNaeiilaWIaem4AaSMaeiilaWIaemOBa4MaeyOeI0Iaem4AaSMaey4kaSIaeGymaeJaeiykaKcabaGae0NSdiMaeiikaGIaem4AaSMaeiilaWIaemOBa4MaeyOeI0Iaem4AaSMaey4kaSIaeGymaeJaeiykaKcaaiaaxMaacaWLjaGaeiikaGIaeG4naCJaeGinaqJaeiykaKcaaa@6AD6@
with (n, k) ∈ ℕ* × ℕ, 0 ≤ k ≤ n and p ∈ [0,1].
Finally, let us remark that the incomplete beta function is differentiable in x and that
∂ β ( x , a , b )  ∂ x   = x a − 1   ( 1 − x )  b − 1         ( 75 )   MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaGGaciab=jGi2kab=j7aIjabcIcaOiabdIha4jabcYcaSiabdggaHjabcYcaSiabdkgaIjabcMcaPaqaaiab=jGi2kabdIha4baacqGH9aqpcqWG4baEdaahaaWcbeqaaiabdggaHjabgkHiTiabigdaXaaakiabcIcaOiabigdaXiabgkHiTiabdIha4jabcMcaPmaaCaaaleqabaGaemOyaiMaeyOeI0IaeGymaedaaOGaaCzcaiaaxMaacqGGOaakcqaI3aWncqaI1aqncqGGPaqkaaa@4D4F@