The gamma function $\Gamma(\alpha)$ is part of the constant term at the front of the [[gamma distribution]]. Note that this function is parameterized with just $\alpha$ (as opposed to the gamma distribution which is parameterized with both $\alpha$ and $\beta$.) The gamma function is defined as $\Gamma(\alpha) = \int_0^\infty x^{\alpha - 1} e^{-x} dx$ ## in the gamma distribution Using a u-substitution ($u = \beta x$), you can find that this integral is equivalent to the integral $\Gamma(\alpha) = \int_0^\infty \beta^\alpha x^{\alpha - 1} e^{-\beta x} dx$ which you'll notice is the rest of the gamma distribution pdf. The trick here is that the gamma function is precisely what we need to make the pdf of the gamma distribution sum to $1$. You can take any non-negative integrable function, integrate it, call it a constant, and put $1$ over your constant in front of any function to create a pdf. # Properties The gamma function of $1$ is $1$. $\Gamma(1) = 1$ The Gamma Function has a recursive property which can be used to "bring the function down" (when $\alpha > 0$) $\Gamma(\alpha) = (\alpha - 1)\Gamma(\alpha - 1)$ This can be extended to show that $\Gamma(n) = (n-1)!$