Editing Beta distribution (section)

=====Moments of linearly transformed, product and inverted random variables=====
One can also show the following expectations for a transformed random variable,<ref name=JKB/> where the random variable ''X'' is Beta-distributed with parameters ''α'' and ''β'': ''X'' ~ Beta(''α'',&nbsp;''β'').  The expected value of the variable 1&nbsp;−&nbsp;''X'' is the mirror-symmetry of the expected value based on&nbsp;''X'':

:<math>\begin{align}
\operatorname{E}[1-X] &= \frac{\beta}{\alpha + \beta} \\
\operatorname{E}[X(1-X)] &= \operatorname{E}[(1-X)X] = \frac{\alpha\beta}{(\alpha+\beta)(\alpha+\beta+1)}
\end{align}</math>

Due to the mirror-symmetry of the probability density function of the beta distribution, the variances based on variables ''X'' and 1&nbsp;−&nbsp;''X'' are identical, and the covariance on ''X''(1&nbsp;−&nbsp;''X'' is the negative of the variance:

:<math>\operatorname{var}[(1-X)]=\operatorname{var}[X] = -\operatorname{cov}[X,(1-X)]= \frac{\alpha \beta}{(\alpha + \beta)^2(\alpha + \beta + 1)}</math>

These are the expected values for inverted variables, (these are related to the harmonic means, see {{section link||Harmonic mean}}):

:<math>\begin{align}
\operatorname{E} \left [\frac{1}{X} \right ] &= \frac{\alpha+\beta-1  }{\alpha -1 } && \text{ if  } \alpha > 1\\
\operatorname{E}\left [\frac{1}{1-X} \right ] &=\frac{\alpha+\beta-1  }{\beta-1 } && \text{ if } \beta > 1
\end{align}</math>

The following transformation by dividing the variable ''X'' by its mirror-image ''X''/(1&nbsp;−&nbsp;''X'') results in the expected value of the "inverted beta distribution" or [[beta prime distribution]] (also known as beta distribution of the second kind or [[Pearson distribution|Pearson's Type VI]]):<ref name=JKB/>

:<math> \begin{align}
\operatorname{E}\left[\frac{X}{1-X}\right] &=\frac{\alpha}{\beta - 1 } && \text{ if }\beta > 1\\
\operatorname{E}\left[\frac{1-X}{X}\right] &=\frac{\beta}{\alpha- 1 } && \text{ if }\alpha > 1
\end{align} </math>

Variances of these transformed variables can be obtained by integration, as the expected values of the second moments centered on the corresponding variables:

:<math>\operatorname{var} \left[\frac{1}{X} \right] =\operatorname{E}\left[\left(\frac{1}{X} - \operatorname{E}\left[\frac{1}{X} \right ] \right )^2\right]= \operatorname{var}\left [\frac{1-X}{X} \right ] =\operatorname{E} \left [\left (\frac{1-X}{X} - \operatorname{E}\left [\frac{1-X}{X} \right ] \right )^2 \right ]= \frac{\beta (\alpha+\beta-1)}{(\alpha -2)(\alpha-1)^2 } \text{ if }\alpha > 2</math>

The following variance of the variable ''X'' divided by its mirror-image (''X''/(1−''X'') results in the variance of the "inverted beta distribution" or [[beta prime distribution]] (also known as beta distribution of the second kind or [[Pearson distribution|Pearson's Type VI]]):<ref name=JKB/>

:<math>\operatorname{var} \left [\frac{1}{1-X} \right ] =\operatorname{E} \left [\left(\frac{1}{1-X} - \operatorname{E} \left [\frac{1}{1-X} \right ] \right)^2 \right ]=\operatorname{var} \left [\frac{X}{1-X} \right ] = \operatorname{E} \left [\left (\frac{X}{1-X} - \operatorname{E} \left [\frac{X}{1-X} \right ] \right )^2 \right ]= \frac{\alpha(\alpha+\beta-1)}{(\beta-2)(\beta-1)^2 } \text{ if }\beta > 2</math>

The covariances are:

:<math>\operatorname{cov}\left [\frac{1}{X},\frac{1}{1-X} \right ] = \operatorname{cov}\left[\frac{1-X}{X},\frac{X}{1-X} \right] =\operatorname{cov}\left[\frac{1}{X},\frac{X}{1-X}\right ] = \operatorname{cov}\left[\frac{1-X}{X},\frac{1}{1-X} \right] =\frac{\alpha+\beta-1}{(\alpha-1)(\beta-1) } \text{ if } \alpha, \beta > 1</math>
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These expectations and variances appear in the four-parameter Fisher information matrix ({{section link||Fisher information}}.)