CÁLCULO GRACELI SINTÉTICO.

$G_{X}=e^{\operatorname {E} [\ln X]}=e^{\psi (\alpha )-\psi (\alpha +\beta )}$ / $f(r,\theta ,\phi )=R(r)Y(\theta ,\varphi )$ $f [$ $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$

onde $\psi$ é a função digama.

Assim sendo, a média geométrica de uma distribuição beta com parâmetros $\alpha$ e $\beta$ é a exponencial da função digama de $\alpha$ e $\beta$ como segue:

{\begin{aligned}F(x;\alpha ,\beta )&={\frac {\mathrm {B} {}(x;\alpha ,\beta )}{\mathrm {B} {}(\alpha ,\beta )}}\\\\&={\frac {\int _{0}^{x}t^{a-1}\,(1-t)^{b-1}\,dt}{B(\alpha ,\beta )}}\\\\&=I_{x}(\alpha ,\beta )\end{aligned}}

/

Mo(X)={\frac {\alpha -1}{\alpha +\beta -2}}

/

G_{X}=e^{\operatorname {E} [\ln X]}=e^{\psi (\alpha )-\psi (\alpha +\beta )}

/

f(r,\theta ,\phi )=R(r)Y(\theta ,\varphi )

f [

\lambda

ω

{\mathcal {T}}

,

\sigma

,

\psi

\mathrm {B} (\alpha ,\beta )={\frac {\Gamma (\alpha )\Gamma (\beta )}{\Gamma (\alpha +\beta )}}

/

Mo(X)={\frac {\alpha -1}{\alpha +\beta -2}}

/

G_{X}=e^{\operatorname {E} [\ln X]}=e^{\psi (\alpha )-\psi (\alpha +\beta )}

/

f(r,\theta ,\phi )=R(r)Y(\theta ,\varphi )

f [

\lambda

ω

{\mathcal {T}}

,

\sigma

,

\psi

1+\sum _{k=1}^{\infty }\left(\prod _{r=0}^{k-1}{\frac {\alpha +r}{\alpha +\beta +r}}\right){\frac {t^{k}}{k!}}

/

Mo(X)={\frac {\alpha -1}{\alpha +\beta -2}}

/

G_{X}=e^{\operatorname {E} [\ln X]}=e^{\psi (\alpha )-\psi (\alpha +\beta )}

/

f(r,\theta ,\phi )=R(r)Y(\theta ,\varphi )

f [

\lambda

ω

{\mathcal {T}}

,

\sigma

,

\psi

G_{X}=e^{\operatorname {E} [\ln X]}=e^{\psi (\alpha )-\psi (\alpha +\beta )}

T^{\nu }{}_{\alpha }

T^{\nu }{}_{\alpha }

T^{\nu }{}_{\alpha }

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