https://www.cfd-online.com/W/index.php?title=Special:Contributions/Hannes79&feed=atom&limit=50&target=Hannes79&year=&month=CFD-Wiki - User contributions [en]2024-03-19T12:00:50ZFrom CFD-WikiMediaWiki 1.16.5https://www.cfd-online.com/Wiki/Beta_PDFBeta PDF2007-07-27T09:00:08Z<p>Hannes79: </p>
<hr />
<div>A <math> \beta </math> [[probability density function]] depends on<br />
two moments only; the mean <math> \mu </math> and the variance <math> \sigma </math>.<br />
This function is widely used in turbulent combustion to define the scalar distribution at each<br />
computational point as a function of the mean and variance.<br />
Assuming that the sample space of the scalar varies betwen 0 and 1.<br />
The beta function PDF has the form<br />
:<math><br />
P (\eta) = \frac{\eta^{\alpha-1} (1- \eta)^{\beta-1}}{\Gamma(\alpha) \Gamma(\beta)}<br />
\Gamma(\alpha + \beta)<br />
</math><br />
where <math> \Gamma </math> is the gamma function and the parameters <br />
<math>\alpha</math> and <math> \beta </math> are related through<br />
<br />
:<math><br />
\alpha = \mu \gamma<br />
</math><br />
<br />
:<math><br />
\beta = (1- \mu) \gamma<br />
</math><br />
<br />
where <math> \gamma</math> is<br />
<br />
:<math><br />
\gamma = \frac{\mu (1- \mu)}{\sigma} -1 <br />
</math></div>Hannes79https://www.cfd-online.com/Wiki/Beta_PDFBeta PDF2007-07-27T08:59:36Z<p>Hannes79: </p>
<hr />
<div>A <math> \beta </math> [[probability density function]] depends on<br />
two moments only; the mean <math> \mu </math> and the variance <math> \sigma </math>.<br />
This function is widely used in turbulent combustion to define the scalar distribution at each<br />
computational point as a function of the mean and variance.<br />
Assuming that the sample space of the scalar varies betwen 0 and 1.<br />
The beta function PDF has the form<br />
:<math><br />
P (\eta) = \frac{\eta^\{alpha-1} (1- \eta)^{\beta-1}}{\Gamma(\alpha) \Gamma(\beta)}<br />
\Gamma(\alpha + \beta)<br />
</math><br />
where <math> \Gamma </math> is the gamma function and the parameters <br />
<math>\alpha</math> and <math> \beta </math> are related through<br />
<br />
:<math><br />
\alpha = \mu \gamma<br />
</math><br />
<br />
:<math><br />
\beta = (1- \mu) \gamma<br />
</math><br />
<br />
where <math> \gamma</math> is<br />
<br />
:<math><br />
\gamma = \frac{\mu (1- \mu)}{\sigma} -1 <br />
</math></div>Hannes79