next up previous
Next: Noise Model MCMC Sampling Up: appendix Previous: appendix

Gamma Distribution

$ x$ has a two-parameter gamma distribution, denoted by $ Ga(a,b)$, with parameters $ a$ and $ b$, if its density is given by:

$\displaystyle f_{Ga}(x;a, b)=\frac{b^a}{\Gamma(a)}x^{a-1}e^{-bx}$ (27)

where $ \Gamma(a)$ is the Gamma function. A $ \chi^2$ distribution with $ \nu$ degrees of freedom corresponds to the distribution $ Ga(\nu/2,1/2)$. The $ b$ parameter is a scale parameter. The one-parameter gamma distribution corresponds to $ Ga(a,1)$. A sample from $ Ga(a,b)$ can be obtained by taking a sample from $ Ga(a,1)$ and dividing it by $ b$. Note, that a gamma distribution has $ mean=a/b$ and $ variance=a/b^2$