# Beta PDF

A $\beta$ probability density function depends on two moments only; the mean $\mu$ and the variance $\sigma$. This function is widely used in turbulent combustion to define the scalar distribution at each computational point as a function of the mean and variance. Assuming that the sample space of the scalar varies betwen 0 and 1. The beta function PDF has the form

$P (\eta) = \frac{\eta^{\alpha-1} (1- \eta)^{\beta-1}}{\Gamma(\alpha) \Gamma(\beta)} \Gamma(\alpha + \beta)$

where $\Gamma$ is the gamma function and the parameters $\alpha$ and $\beta$ are related through

$\alpha = \mu \gamma$
$\beta = (1- \mu) \gamma$

where $\gamma$ is

$\gamma = \frac{\mu (1- \mu)}{\sigma} -1$