Some Gamma Function Notes

The Gamma Function is a particular form of integral that is commonly seen in probability problems:
$\Gamma(\alpha) = \int_{0}^{\infty}x^{\alpha - 1}e^{-x}dx$

The Gamma Function is computed as a factorial if $\alpha$ is an integer:
$\Gamma(\alpha) = (\alpha - 1)!$

The exponent in the integral can be different:
$\Gamma(\alpha + 1) = \alpha\Gamma(\alpha) = \int_{0}^{\infty}x^{\alpha}e^{-x}dx$

Or in general:
$\Gamma(\alpha + n) = (\alpha+n-1)(\alpha+n-2)...(\alpha+1)\alpha\Gamma(\alpha)$