The binomial distribution

is rearranged in the general exponential form using the same trick as was used for the Bernoulli distribution, as f (y, n) = exp{[y log(x/(1 — ¡x)) + nlog(1 — ¡)]n + log[n!/((ny)!(n — ny)!)]}.

This again gives 9 = log(x/(1 — ¡x)) and the logit as the canonical link function. Therefore, ¡x can again be expressed as exp(9)/(1 + exp(9)) = (1 + exp(—9))—1 and we can write f (y, n) = exp{[y9 + log{1 — exp(9)/(1 + exp(9))}]n + log[n!/((ny)!(n — ny)!)]} = exp{[y9 — log( 1 + exp(9))]n + log[n!/((ny)!(n — ny)!)]}.

Therefore, b(9) = log(1 + exp(9)), a = 1/n, and c(y) = log[n!/((ny)!(n — ny)!)].

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