The citation is by combining pages 523 and 461 of

‘Distributions for Modeling Location, Scale, and Shape: Using GAMLSS in R’ R. A. Rigby, M. D. Stasinopoulos, G. Z. Heller and F. De Bastiani. Chapman and Hall/CRC, Boca Raton, 2019

Page 523 says if Y ~ BB(n, mu, sigma), then Y|p ~ BI(n,p) where p ~ BEo(mu/sigma, [1-mu]/sigma) i.e. p ~ BEo(a, b) where a=mu/sigma and b = [1-mu]/sigma.

Page 461 says BEo(a, b) = BE(mu_1,sigma_1) where mu_1 = a/(a+b) and sigma_1 = (a+b+1)^(-0.5).

So mu_1 = mu and sigma_1 = [sigma/(1+sigma)]^0.5.

So in conclusion if

Y ~ BB(n, mu, sigma), then

Y|p ~ BI(n,p) where
p ~ BE(mu, [sigma/(1+sigma)]^0.5).

The citation is by combining pages 523 and 461 of

‘Distributions for Modeling Location, Scale, and Shape: Using GAMLSS in R’ R. A. Rigby, M. D. Stasinopoulos, G. Z. Heller and F. De Bastiani. Chapman and Hall/CRC, Boca Raton, 2019

Page 523 says if Y ~ BB(n, mu, sigma), then Y|p ~ BI(n,p) where p ~ BEo(mu/sigma, [1-mu]/sigma)

i.e. p ~ BEo(a, b) where a=mu/sigma and b = [1-mu]/sigma.

Page 461 says BEo(a, b) = BE(mu_1,sigma_1) where mu_1 = a/(a+b) and sigma_1 = (a+b+1)^(-0.5).

So mu_1 = mu and sigma_1 = [sigma/(1+sigma)]^0.5.

So in conclusion if

Y ~ BB(n, mu, sigma), then

Y|p ~ BI(n,p) where
p ~ BE(mu, [sigma/(1+sigma)]^0.5).