# Beta-Binomial Derivation

I was looking on wikipedia and seemed to be not follow how the beta-binomial is derived. I was hoping I could provide an example I found in a paper and someone could explain the steps.

Given the marginal

$$m(y)=\int_{0}^{1}6{n\choose{y}}p^{y+1}(1-p)^{n-y+1}dp$$

How do I get to here

$$=6{n\choose{y}}\frac{\Gamma(y+2)\Gamma(n-y+2)}{\Gamma(n+4)}$$

• are you familiar with the beta function and its relationship with the gamma function? – jld Nov 11 '17 at 1:47
• That is what I am confused about. Any information or link would be helpful. Thanks! – Alex Nov 11 '17 at 1:56
• If you click the link (in wikipedia) on the gamma and/or beta function there is explanation there of how they are related – probabilityislogic Nov 11 '17 at 2:38

## 1 Answer

There are just two things you need to know for this.

First one: definition of the Beta function.

$$B(x,y) := \int_0^1 t^{x-1}(1-t)^{y-1}\,\text dt.$$

Second thing: the identity that $$B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$$ (see here for a proof). So you're basically there already, we just tidy up by hiding the integral in some nice special functions.