Gamma Function

Definition

The Gamma function extends the factorial to real (and complex) numbers:

$$ \Gamma(z) = \int_0^{\infty} t^{z-1} e^{-t}\, dt, \quad z > 0. $$

Key Properties

  • $\Gamma(n) = (n-1)!$ for positive integers $n$.
  • $\Gamma(z+1) = z\,\Gamma(z)$ (recursive property).
  • $\Gamma(1/2) = \sqrt{\pi}$.

Role in Statistics

The Gamma function appears in the normalizing constants of the Beta Distribution, Weibull Distribution, Gamma distribution, and Student-$t$ distribution.