cdf_BB(i, sample_size, beta_shape1, beta_shape2) = sf_BNB (sample_size – i - 1, i + 1, beta_shape1, beta_shape2)

sf_BB(i, sample_size, beta_shape1, beta_shape2) = sf_BNB (i, sample_size - i, beta_shape2, beta_shape2).

sf_BNB(i,r,a,b) = pmf_BNB(i+1,r,a,b) + sf_BNB(i+1,r,a,b)

sf_BNB(i,r,a,b) = PBB(i+1,r-1,b,a) + sf_BNB(i+1,r-1,a,b)

sf_BNB(i,r,a,b) = PBB(i+1,b-1,r,a) + sf_BNB(i+1,r,a,b-1)

sf_BNB(i,r,a,b) = HT(i+1,r-1,b-1,a) + sf_BNB(i+1,r-1,a+1,b-1)

sf_BNB(i,r,a,b) = PBB(a,r-1,b, i+1) + sf_BNB(i,r-1,a+1,b)

sf_BNB(i,r,a,b) = PBB(a,b-1,r, i+1) + sf_BNB(i,r,a+1,b-1)

sf_BNB(i,r,a,b) = pBNB(a,r,i+1,b) + sf_BNB(i,r,a+1,b)

Pr(X=i;n,k,m)
= (m-n)!n!(m-k)!k!/i!/(n-i)!/(k-i)!/(m-k-n+i)!/m! if i >= 0 and i <= n and i <= k and k+n <= m+i
= 0 otherwise.