I am reviewing some statistics problems before my exam and I have stumbled upon a calculation that I am having trouble wrapping my head around. Could someone briefly explain to me how my professor went from line (2) to line (3) using the gamma function? Since the gamma function involves an integral, I am struggling to see the trick to go from line (2) to line (3). enter image description here
2026-03-27 09:37:56.1774604276
Gamma and Beta distributions
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To add a touch more detail to what was written by Henry in the comments, you can use integration by parts to show that $\Gamma(x+1) = x \cdot \Gamma(x)$ for any positive real number $x$. You can then inductively expand this out to get the identity mentioned in Henry's comment and see that there are four groups of expressions collected in such a way (two in the numerators and two in the denominators).