The Continuous Binomial distribution

If we examine the probability mass function of a binomial distribution, we have

$${\rm p}(x;n,p) = {n \choose x} p^x (1-p)^{n-x} \hbox{~~~for ~} x=0,1,2,\dots,n$$ (1)

There are a few problems involved with applying this distribution to our model. First, we do not know the appropriate value for n. Next, we are no longer dealing with discrete parameters, so n is a continuous parameter. Let us construct a new ``continuous binomial'' distribution by first noticing that the binomial coefficient can be expressed in terms of Gamma functions, which are continuous in nature.

$${n \choose x} = {{n!} \over {x! (n-x)!}} = {{\Gamma(n+1)} \over {\Gamma(x+1) \Gamma(n-x+1)}}$$ (2)

Hence, equation 1 is equivalent to the continuous binomial

$$f(x;n,p) = {{\Gamma(n+1)} \over {\Gamma(x+1) \Gamma(n-x+1)}} p^x (1-p)^{n-x} \hbox{~~~for~} 0 \leq x \leq n$$ (3)