Editing Beta distribution (section)

====Four parameters====
A beta distribution with the two shape parameters ''α'' and ''β'' is supported on the range [0,1] or (0,1).  It is possible to alter the location and scale of the distribution by introducing two further parameters representing the minimum, ''a'', and maximum ''c'' (''c'' > ''a''), values of the distribution,<ref name=JKB/>  by a linear transformation substituting the non-dimensional variable ''x'' in terms of the new variable ''y'' (with support [''a'',''c''] or (''a'',''c'')) and the parameters ''a'' and ''c'':

:<math>y = x(c-a) + a,  \text{ therefore } x = \frac{y-a}{c-a}.</math>

The [[probability density function]] of the four parameter beta distribution is equal to the two parameter distribution, scaled by the range (''c''&nbsp;−&nbsp;''a''), (so that the total area under the density curve equals a probability of one), and with the "y" variable shifted and scaled as follows:
::<math>f(y; \alpha, \beta, a, c) = \frac{f(x;\alpha,\beta)}{c-a} =\frac{\left(\frac{y-a}{c-a}\right)^{\alpha-1} \left (\frac{c-y}{c-a} \right)^{\beta-1} }{(c-a)B(\alpha, \beta)}=\frac{ (y-a)^{\alpha-1} (c-y)^{\beta-1} }{(c-a)^{\alpha+\beta-1}B(\alpha, \beta)}.</math>

That a random variable ''Y'' is beta-distributed with four parameters ''α'', ''β'', ''a'', and ''c'' will be denoted by:

:<math>Y \sim \operatorname{Beta}(\alpha, \beta, a, c).</math>

Some measures of central location are scaled (by (''c''&nbsp;−&nbsp;''a'')) and shifted (by ''a''), as follows:

:<math> \begin{align}
\mu_Y &= \mu_X(c-a) + a \\
& = \left(\frac{\alpha}{\alpha+\beta}\right)(c-a) + a = \frac{\alpha c+ \beta a}{\alpha+\beta} \\[8pt]
\text{mode}(Y) &=\text{mode}(X)(c-a) + a \\
& = \left(\frac{\alpha - 1}{\alpha+\beta - 2}\right)(c-a) + a = \frac{(\alpha-1) c+(\beta-1) a}{\alpha+\beta-2}\ ,\qquad \text{ if } \alpha, \beta>1 \\[8pt]
\text{median}(Y) &= \text{median}(X)(c-a) + a \\
& = \left (I_{\frac{1}{2}}^{[-1]}(\alpha,\beta) \right )(c-a)+a
\end{align}
</math>

Note: the geometric mean and harmonic mean cannot be transformed by a linear transformation in the way that the mean, median and mode can.

The shape parameters of ''Y'' can be written in term of its mean and variance as

:<math> \begin{align}
\alpha &= \frac{(a - \mu_Y)(a \, c - a \, \mu_Y - c \, \mu_Y + \mu_Y^2 + \sigma_Y^2)}{\sigma_Y^2(c-a)} \\
\beta &=  -\frac{(c - \mu_Y)(a \, c - a \, \mu_Y - c \, \mu_Y + \mu_Y^2 + \sigma_Y^2)}{\sigma_Y^2(c-a)}
\end{align}
</math>

The statistical dispersion measures are scaled (they do not need to be shifted because they are already centered on the mean) by the range (''c''&nbsp;−&nbsp;''a''), linearly for the mean deviation and nonlinearly for the variance:

::<math>\text{(mean deviation around mean)}(Y)=</math>
::<math>(\text{(mean deviation around mean)}(X))(c-a) =\frac{2 \alpha^\alpha \beta^\beta}{\Beta(\alpha,\beta)(\alpha + \beta)^{\alpha + \beta + 1}}(c-a)</math>
::<math> \text{var}(Y) =\text{var}(X)(c-a)^2 =\frac{\alpha\beta (c-a)^2}{(\alpha+\beta)^2(\alpha+\beta+1)}.</math>

Since the [[skewness]] and [[excess kurtosis]] are non-dimensional quantities (as [[Moment (mathematics)|moments]] centered on the mean and normalized by the [[standard deviation]]), they are independent of the parameters ''a'' and ''c'', and therefore equal to the expressions given above in terms of ''X'' (with support [0,1] or (0,1)):

::<math> \text{skewness}(Y) =\text{skewness}(X) = \frac{2 (\beta - \alpha) \sqrt{\alpha + \beta + 1} }{(\alpha + \beta + 2) \sqrt{\alpha \beta}}.</math>

::<math> \text{kurtosis excess}(Y) =\text{kurtosis excess}(X)=\frac{6[(\alpha - \beta)^2 (\alpha +\beta + 1) - \alpha \beta (\alpha + \beta + 2)]}
{\alpha \beta (\alpha + \beta + 2) (\alpha + \beta + 3)} </math>