Editing Beta distribution (section)

=====Mean and variance=====

Solving the system of (coupled) equations given in the above sections as the equations for the mean and the variance of the beta distribution in terms of the original parameters ''α'' and ''β'', one can express the ''α'' and ''β'' parameters in terms of the mean (''μ'') and the variance (var):

:<math> \begin{align}
\nu &= \alpha + \beta = \frac{\mu(1-\mu)}{\mathrm{var}}-1, \text{ where }\nu =(\alpha + \beta)  >0,\text{ therefore: }\text{var}< \mu(1-\mu)\\
\alpha&= \mu \nu =\mu \left(\frac{\mu(1-\mu)}{\text{var}}-1\right), \text{ if } \text{var}< \mu(1-\mu)\\
\beta &= (1 - \mu) \nu = (1 - \mu)\left(\frac{\mu(1-\mu)}{\text{var}}-1\right), \text{ if }\text{var}< \mu(1-\mu).
\end{align}</math>

This [[Statistical parameter|parametrization]] of the beta distribution may lead to a more intuitive understanding than the one based on the original parameters ''α'' and ''β''. For example, by expressing the mode, skewness, excess kurtosis and differential entropy in terms of the mean and the variance:

[[File:Mode Beta Distribution for both alpha and beta greater than 1 - J. Rodal.jpg|325px]]
[[File:Mode Beta Distribution for both alpha and beta greater than 1 - another view - J. Rodal.jpg|325px]]
[[File:Skewness Beta Distribution for mean full range and variance between 0.05 and 0.25 - Dr. J. Rodal.jpg|325px]]
[[File:Skewness Beta Distribution for mean and variance both full range - J. Rodal.jpg|325px]]
[[File:Excess Kurtosis Beta Distribution with mean for full range and variance from 0.05 to 0.25 - J. Rodal.jpg|325px]]
[[File:Excess Kurtosis Beta Distribution with mean and variance for full range - J. Rodal.jpg|325px]]
[[File:Differential Entropy Beta Distribution with mean from 0.2 to 0.8 and variance from 0.01 to 0.09 - J. Rodal.jpg|325px]]
[[File:Differential Entropy Beta Distribution with mean from 0.3 to 0.7 and variance from 0 to 0.2 - J. Rodal.jpg|325px]]