Gamma Distribution

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

$$\mathcal{P}(x\vert a, b)=\frac{b^a}{\Gamma(a)}x^{a-1}e^{-bx}$$ (22)

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