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![{\displaystyle \operatorname {E} [\ln X]=\psi (\alpha )-\psi (\alpha +\beta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/350f451e381db241e3b1ec2cc9ba84c2ebf27e93)
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![{\displaystyle \operatorname {var} [X]={\frac {\alpha \beta }{(\alpha +\beta )^{2}(\alpha +\beta +1)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c2367c268f610193119758414345f42b6958e0c)
![{\displaystyle \operatorname {var} [\ln X]=\psi _{1}(\alpha )-\psi _{1}(\alpha +\beta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e396e8700267735eb741f73e8906445579c43bc6)
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![{\displaystyle {\begin{matrix}\ln \mathrm {B} (\alpha ,\beta )-(\alpha -1)\psi (\alpha )-(\beta -1)\psi (\beta )\\[0.5em]+(\alpha +\beta -2)\psi (\alpha +\beta )\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c89e36ccbf7522eba17d6e5ddb267e7cef46b8e)
![{\displaystyle {\begin{aligned}\mu =\operatorname {E} [X]&=\int _{0}^{1}xf(x;\alpha ,\beta )\,dx\\&=\int _{0}^{1}x\,{\frac {x^{\alpha -1}(1-x)^{\beta -1}}{\mathrm {B} (\alpha ,\beta )}}\,dx\\&={\frac {\alpha }{\alpha +\beta }}\\&={\frac {1}{1+{\frac {\beta }{\alpha }}}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9137834d9d47360ed6c23550c6236fed5fd35f7)
![{\displaystyle {\begin{aligned}\operatorname {E} [\ln X]&=\int _{0}^{1}\ln x\,f(x;\alpha ,\beta )\,dx\\[4pt]&=\int _{0}^{1}\ln x\,{\frac {x^{\alpha -1}(1-x)^{\beta -1}}{\mathrm {B} (\alpha ,\beta )}}\,dx\\[4pt]&={\frac {1}{\mathrm {B} (\alpha ,\beta )}}\,\int _{0}^{1}{\frac {\partial x^{\alpha -1}(1-x)^{\beta -1}}{\partial \alpha }}\,dx\\[4pt]&={\frac {1}{\mathrm {B} (\alpha ,\beta )}}{\frac {\partial }{\partial \alpha }}\int _{0}^{1}x^{\alpha -1}(1-x)^{\beta -1}\,dx\\[4pt]&={\frac {1}{\mathrm {B} (\alpha ,\beta )}}{\frac {\partial \mathrm {B} (\alpha ,\beta )}{\partial \alpha }}\\[4pt]&={\frac {\partial \ln \mathrm {B} (\alpha ,\beta )}{\partial \alpha }}\\[4pt]&={\frac {\partial \ln \Gamma (\alpha )}{\partial \alpha }}-{\frac {\partial \ln \Gamma (\alpha +\beta )}{\partial \alpha }}\\[4pt]&=\psi (\alpha )-\psi (\alpha +\beta )\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd9db519e08e3c72cd6f9e2f0c90a7c57bdba035)
![{\displaystyle var(X)=E[x-\mu ^{2}]={\alpha \beta \over (\alpha +\beta )^{2}(\alpha +\beta +1)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84dc373c2f4bcf7a7388d66eb3ccca7bbb0af2aa)
G = SINC [x] = sin [ ζ [s]] / { } / [ ζ [s]] / { } G = SINC [x] = [sin + + . . ] / [ + + . . ]= S [x] = sin [ + + . . ]= C [x] = COS [ + + . . ] = G [ , ζ ] = [ ζ [s]] / { } = G = SINC [x] = [sin + + . . S [x] = sin [ + + . . ]=
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