The Beta prior distribution of the Continuous Binomial

In order to generate a confidence interval for the parameter p, we must first determine the distribution of that parameter. The likelihood equation for p is equal to the continuous binomial density function, but with p as the independent variable and x as a parameter. If we make a change of variables and introduce two new parameters a and b defined by

$$\matrix{ a = np + 1 \cr b = n(1-p) + 1}$$ (4)

then we get the following function for the distribution of p

$$f(p;a,b) = {{\Gamma(a+b-1)} \over {\Gamma(a) \Gamma(b)}} p^{a-1} (1-p)^{b-1} \hbox{~~~for~} 0 \leq p \leq 1$$ (5)

The integral of this function over the range [0,1] is not equal to one. However, if we recall the fact that $\Gamma(a+b-1)=(a+b)\Gamma(a+b)$ then we can divide equation 5 by (a+b) to get

$$f(p;a,b) = {{\Gamma(a+b) \over {\Gamma(a) \Gamma(b)}}} p^{a-1} (1-p)^{b-1} \hbox{~~~for~} 0 \leq p \leq 1$$ (6)

Which is the probability density function for a Beta distribution with parameters a and b. We still do not have enough information to create a confidence interval for p using this distribution because there is no estimate for the binomial parameter n. We do, however, have the variance of the parameter estimate p.