Editing Beta distribution (section)

===Symmetry===
All statements are conditional on ''α'', ''β'' > 0:

* '''Probability density function''' [[Symmetry|reflection symmetry]]
::<math>f(x;\alpha,\beta) = f(1-x;\beta,\alpha)</math>

* '''Cumulative distribution function''' [[Symmetry|reflection symmetry]] plus unitary [[Symmetry|translation]]
::<math>F(x;\alpha,\beta) = I_x(\alpha,\beta) = 1- F(1- x;\beta,\alpha) = 1 - I_{1-x}(\beta,\alpha)</math>

* '''Mode''' [[Symmetry|reflection symmetry]] plus unitary [[Symmetry|translation]]
::<math>\operatorname{mode}(\Beta(\alpha, \beta))= 1-\operatorname{mode}(\Beta(\beta, \alpha)),\text{ if }\Beta(\beta, \alpha)\ne \Beta(1,1)</math>

* '''Median''' [[Symmetry|reflection symmetry]] plus unitary [[Symmetry|translation]]
::<math>\operatorname{median} (\Beta(\alpha, \beta) )= 1 - \operatorname{median} (\Beta(\beta, \alpha))</math>

* '''Mean''' [[Symmetry|reflection symmetry]] plus unitary [[Symmetry|translation]]
::<math>\mu (\Beta(\alpha, \beta) )= 1 - \mu (\Beta(\beta, \alpha) )</math>

* '''Geometric means''' each is individually asymmetric, the following symmetry applies between the geometric mean based on ''X'' and the geometric mean based on its [[Reflection formula|reflection]] (1-X)
::<math>G_X (\Beta(\alpha, \beta) )=G_{(1-X)}(\Beta(\beta, \alpha) ) </math>

* '''Harmonic means''' each is individually asymmetric, the following symmetry applies between the harmonic mean based on ''X'' and the harmonic mean based on its [[Reflection formula|reflection]] (1-X)
::<math>H_X (\Beta(\alpha, \beta) )=H_{(1-X)}(\Beta(\beta, \alpha) ) \text{ if } \alpha, \beta > 1 </math> .

* '''Variance''' symmetry
::<math>\operatorname{var} (\Beta(\alpha, \beta) )=\operatorname{var} (\Beta(\beta, \alpha) )</math>

* '''Geometric variances''' each is individually asymmetric, the following symmetry applies between the log geometric variance based on X and the log geometric variance based on its [[Reflection formula|reflection]] (1-X)
::<math>\ln(\operatorname{var_{GX}} (\Beta(\alpha, \beta))) = \ln(\operatorname{var_{G(1-X)}}(\Beta(\beta, \alpha))) </math>

* '''Geometric covariance''' symmetry
::<math>\ln \operatorname{cov_{GX,(1-X)}}(\Beta(\alpha, \beta))=\ln \operatorname{cov_{GX,(1-X)}}(\Beta(\beta, \alpha))</math>

* '''Mean [[absolute deviation]] around the mean''' symmetry
::<math>\operatorname{E}[|X - E[X]| ] (\Beta(\alpha, \beta))=\operatorname{E}[| X - E[X]|] (\Beta(\beta, \alpha))</math>

* '''Skewness''' [[Symmetry (mathematics)|skew-symmetry]]
::<math>\operatorname{skewness} (\Beta(\alpha, \beta) )= - \operatorname{ skewness} (\Beta(\beta, \alpha) )</math>

* '''Excess kurtosis''' symmetry
::<math>\text{excess kurtosis} (\Beta(\alpha, \beta) )= \text{excess kurtosis} (\Beta(\beta, \alpha) )</math>

* '''Characteristic function''' symmetry of [[Real part]] (with respect to the origin of variable "t")
::<math> \text{Re} [{}_1F_1(\alpha; \alpha+\beta; it) ] = \text{Re} [ {}_1F_1(\alpha; \alpha+\beta; - it)]  </math>

* '''Characteristic function''' [[Symmetry (mathematics)|skew-symmetry]] of [[Imaginary part]] (with respect to the origin of variable "t")
::<math> \text{Im} [{}_1F_1(\alpha; \alpha+\beta; it) ] = - \text{Im} [ {}_1F_1(\alpha; \alpha+\beta; - it) ]  </math>

* '''Characteristic function''' symmetry of [[Absolute value]] (with respect to the origin of variable "t")
::<math> \text{Abs} [ {}_1F_1(\alpha; \alpha+\beta; it) ] = \text{Abs} [ {}_1F_1(\alpha; \alpha+\beta; - it) ]  </math>

* '''Differential entropy''' symmetry
::<math>h(\Beta(\alpha, \beta) )= h(\Beta(\beta, \alpha) )</math>

* '''Relative entropy (also called [[Kullback&ndash;Leibler divergence]])''' symmetry
::<math>D_{\mathrm{KL}}(X_1\parallel X_2) = D_{\mathrm{KL}}(X_2\parallel X_1), \text{ if }h(X_1) = h(X_2)\text{, for (skewed) }\alpha \neq \beta</math>

* '''Fisher information matrix''' symmetry
::<math>{\mathcal{I}}_{i, j} = {\mathcal{I}}_{j, i}</math>