Under-represented pattern
In the case of an under-represented pattern we have
S N  = log ⁡  10   F −  ( N )   with   F −  ( N ) ≜ ℙ μ N  , π N    ( N ≤ N obs   ) .       ( 25  )        MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqaaeqadaaabaGaem4uam1aaSbaaSqaaGqabiab=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@6905@
Using a binomial approximation we get
F −  ( N ) ≃ ℙ ( ℬ ( ℓ h  , P ( N ) ) ≤ N o b s   ) = β −  ( P ( N ) , N o b s   + 1 , ℓ h  − N o b s   )  β ( N o b s   + 1 , ℓ h  − N o b s   )         ( 26 )   MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaWaaeGaeaaakeaacqWGgbGrdaahaaWcbeqaaiabgkHiTaaakiabcIcaOGqabiab=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@8F8B@
and, by the same method than in the over-represented case we finally have
σ ≃ Q −    t  G × C × G         ( 27  )    MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFdpWCcqWIdjYocqWGrbqudaahaaWcbeqaaiabgkHiTaaakmaakaaabaGaeeiiaaYaaWbaaSqabeaaieGacqGF0baDaaacbeGccqqFhbWrcqGHxdaTcqqFdbWqcqGHxdaTcqqFhbWraSqabaGccaWLjaGaaCzcamaabmaabaGaeGOmaiJaeG4naCdacaGLOaGaayzkaaaaaa@40AE@
where
Q −  = p N obs   + 1   ( 1 − p )  ℓ h  − N obs   − 1    ln ⁡ ( 10 ) β −  ( p , N obs   + 1 , ℓ h  − N obs   )         ( 28  )    MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGrbqudaahaaWcbeqaaiabgkHiTaaakiabg2da9maalaaabaGaemiCaa3aaWbaaSqabeaacqWGobGtdaWgaaadbaGaee4Ba8MaeeOyaiMaee4CamhabeaaliabgUcaRiabigdaXaaakiabcIcaOiabigdaXiabgkHiTiabdchaWjabcMcaPmaaCaaaleqabaGaeS4eHW2aaSbaaWqaaiabdIgaObqabaWccqGHsislcqWGobGtdaWgaaadbaGaee4Ba8MaeeOyaiMaee4CamhabeaaliabgkHiTiabigdaXaaaaOqaaiGbcYgaSjabc6gaUjabcIcaOiabigdaXiabicdaWiabcMcaPGGaciab=j7aInaaCaaaleqabaGaeyOeI0caaOGaeiikaGIaemiCaaNaeiilaWIaemOta40aaSbaaSqaaiabb+gaVjabbkgaIjabbohaZbqabaGccqGHRaWkcqaIXaqmcqGGSaalcqWItecBdaWgaaWcbaGaemiAaGgabeaakiabgkHiTiabd6eaonaaBaaaleaacqqGVbWBcqqGIbGycqqGZbWCaeqaaOGaeiykaKcaaiaaxMaacaWLjaWaaeWaaeaacqaIYaGmcqaI4aaoaiaawIcacaGLPaaaaaa@6C5B@