e-Statistics > 4470-5470 Probability and Statistics I

Gamma distribution

The gamma density is defined as

$\displaystyle f(x) = \frac{\lambda^\alpha}{\Gamma(\alpha)}
x^{\alpha -1}e^{-\lambda x}
\quad x \ge 0

which depends on two parameters $ \alpha > 0$ and $ \lambda > 0$. We call the parameter $ \alpha =$ a shape parameter, because changing $ \alpha$ changes the shape of the density. We call the parameter $ \lambda =$ a rate parameter, which rescales the density without changing its shape. This is equivalent to changing the units of measurement (feet to meters, or seconds to minutes).

The gamma function $ \Gamma(\alpha)$ can be numerically evaluated by

$ \displaystyle
\Gamma(\alpha) = \int_0^\infty u^{\alpha-1}e^{-u} du$ =
However, if $ \alpha$ is an integer $ n$ then $ \Gamma(n) = (n-1)!$.