Editing Beta distribution (section)

=====Two unknown parameters=====
Two unknown parameters (<math> (\hat{\alpha}, \hat{\beta})</math>  of a beta distribution supported in the [0,1] interval) can be estimated, using the method of moments, with the first two moments (sample mean and sample variance) as follows.  Let:

: <math>\text{sample mean(X)}=\bar{x} = \frac{1}{N}\sum_{i=1}^N X_i</math>

be the [[sample mean]] estimate and

: <math>\text{sample variance(X)} =\bar{v} = \frac{1}{N-1}\sum_{i=1}^N (X_i - \bar{x})^2</math>

be the [[sample variance]] estimate.  The [[method of moments (statistics)|method-of-moments]] estimates of the parameters are

:<math>\hat{\alpha} = \bar{x} \left(\frac{\bar{x} (1 - \bar{x})}{\bar{v}} - 1 \right),</math> if <math>\bar{v} <\bar{x}(1 - \bar{x}),</math>
: <math>\hat{\beta} = (1-\bar{x}) \left(\frac{\bar{x} (1 - \bar{x})}{\bar{v}} - 1 \right),</math> if <math>\bar{v}<\bar{x}(1 - \bar{x}).</math>
<!--  MLE's should be in this section too.  Maybe I'll be back.... -->

When the distribution is required over a known interval other than [0, 1] with random variable ''X'', say [''a'', ''c''] with random variable ''Y'', then replace <math>\bar{x}</math> with <math>\frac{\bar{y}-a}{c-a},</math> and <math>\bar{v}</math> with <math>\frac{\bar{v_Y}}{(c-a)^2}</math> in the above couple of equations for the shape parameters (see the "Four unknown parameters" section below),<ref>{{Cite web|url=https://www.itl.nist.gov/div898/handbook/eda/section3/eda366h.htm|title=1.3.6.6.17. Beta Distribution|website=www.itl.nist.gov}}</ref> where:

: <math>\text{sample mean(Y)}=\bar{y} = \frac{1}{N}\sum_{i=1}^N Y_i</math>
: <math>\text{sample variance(Y)} = \bar{v_Y} = \frac{1}{N-1}\sum_{i=1}^N (Y_i - \bar{y})^2</math>