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:
\(\Gamma(\alpha + n) = (\alpha+n-1)(\alpha+n-2)...(\alpha+1)\alpha\Gamma(\alpha)\)