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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

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

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

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