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Beta PDF

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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
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