Користувач:BogdanShevchenko/Чернетка

Exponential family

$p(x|\lambda )=C(\lambda )\lambda ^{x}(1-\lambda )^{1-x}$

Where

$C(\lambda )={\begin{cases}2&{\text{if }}\lambda ={\frac {1}{2}}\\{\frac {2\tanh ^{-1}(1-2\lambda )}{1-2\lambda }}&{\text{ otherwise}}\end{cases}}$

let's assume

$\eta =\ln {\frac {\lambda }{1-\lambda }}=-2tanh^{-1}(1-2\lambda )$

if so, for $\lambda \neq 1/2$

$C(\lambda )={\frac {\eta }{e^{\eta }-1}}$

and in exponential family form

$p(x|\eta )=e^{\eta x-\ln {\frac {e^{\eta }-1}{\eta }}}$

for exponential family, moments of distribution could be calculated from deriatives of log normalizer:

First moment (Mean)

$\mu =E[x]={\frac {d}{d\eta }}a(\eta )={\frac {d}{d\eta }}\ln {\frac {e^{\eta }-1}{\eta }}={\frac {(\eta -1)e^{\eta }+1}{\eta (e^{\eta }-1)}}={\frac {\eta e^{\eta }}{\eta (e^{\eta }-1)}}-{\frac {e^{\eta }-1}{\eta (e^{\eta }-1)}}={\frac {e^{\eta }}{e^{\eta }-1}}-{\frac {1}{\eta }}={\frac {\lambda }{2\lambda -1}}+{\frac {1}{2tanh^{-1}(1-2\lambda )}}$

Second moment (Variance)

$\operatorname {Var} (x)=\mu _{2}=\sigma ^{2}=E[(x-\mu )^{2}]={\frac {d^{2}}{d\eta ^{2}}}\ln {\frac {e^{\eta }-1}{\eta }}=-{\frac {e^{2\eta }}{(e^{\eta }-1)^{2}}}+{\frac {e^{\eta }}{e^{\eta }-1}}+{\frac {1}{\eta ^{2}}}=-\left({\frac {\lambda }{1-\lambda }}{\frac {1-\lambda }{2\lambda -1}}\right)^{2}+{\frac {\lambda }{2\lambda -1}}+{\frac {1}{(2tanh^{-1}(1-2\lambda ))^{2}}}=$

$={\frac {\lambda (2\lambda -1)-\lambda ^{2}}{(2\lambda -1)^{2}}}+{\frac {1}{(2tanh^{-1}(1-2\lambda ))^{2}}}={\frac {\lambda (\lambda -1)}{(2\lambda -1)^{2}}}+{\frac {1}{(2tanh^{-1}(1-2\lambda ))^{2}}}$

Third moment (Skewness)

$\mu _{3}=E[(x-\mu )^{3}]={\frac {d^{3}}{d\eta ^{3}}}\ln {\frac {e^{\eta }-1}{\eta }}={\frac {2e^{3\eta }}{(e^{\eta }-1)^{3}}}-{\frac {3e^{2\eta }}{(e^{\eta }-1)^{2}}}+{\frac {e^{\eta }}{e^{\eta }-1}}-{\frac {2}{\eta ^{3}}}=2\left({\frac {\lambda }{2\lambda -1}}\right)^{3}-3\left({\frac {\lambda }{2\lambda -1}}\right)^{2}+{\frac {\lambda }{2\lambda -1}}+{\frac {2}{(2tanh^{-1}(1-2\lambda ))^{3}}}=$

$={\frac {2\lambda ^{3}-3\lambda ^{2}(2\lambda -1)+(2\lambda -1)^{2}\lambda }{(2\lambda -1)^{3}}}-{\frac {2}{(2tanh^{-1}(1-2\lambda ))^{3}}}={\frac {\lambda (1-\lambda )}{(2\lambda -1)^{3}}}+{\frac {2}{(2tanh^{-1}(1-2\lambda ))^{3}}}$

Skewness is equal to:

$\operatorname {Skew} [X]={\frac {\mu _{3}}{\sigma ^{3}}}={\frac {\mu _{3}}{\mu _{2}^{3/2}}}=-{\frac {\eta ^{3}\lambda ^{2}-\eta ^{3}\lambda +16\lambda ^{3}-24\lambda ^{2}+12\lambda -2}{\eta ^{3}(2\lambda -1)^{3}\left({\frac {1}{\eta ^{2}}}+{\frac {\lambda (1-\lambda )}{2\lambda -1}}\right)^{3/2}}}$

Fourth moment (Kurtosis)

$\mu _{4}=E[(x-\mu )^{4}]={\frac {d^{4}}{d\eta ^{4}}}\ln {\frac {e^{\eta }-1}{\eta }}=-{\frac {6e^{3\eta }}{(e^{\eta }-1)^{4}}}+{\frac {6e^{2\eta }}{(e^{\eta }-1)^{3}}}-{\frac {e^{\eta }}{(e^{\eta }-1)^{2}}}+{\frac {6}{\eta ^{4}}}=$

$={\frac {-6\lambda ^{3}(1-\lambda )+6\lambda ^{2}(1-\lambda )(2\lambda -1)-\lambda (1-\lambda )(2\lambda -1)^{2}}{(2\lambda -1)^{4}}}+{\frac {6}{(2tanh^{-1}(1-2\lambda ))^{4}}}={\frac {\lambda (1-\lambda )(2\lambda ^{2}-2\lambda -1)}{(2\lambda -1)^{4}}}+{\frac {6}{(2tanh^{-1}(1-2\lambda ))^{4}}}$

Kurtosis is equal to:

$\operatorname {Kurt} [X]={\frac {\mu _{4}}{\sigma ^{4}}}={\frac {\mu _{4}}{\mu _{2}^{2}}}={\frac {-2\lambda ^{4}\eta ^{4}+4\eta ^{4}\lambda ^{3}-\eta ^{4}\lambda ^{2}-\eta ^{4}\lambda +96\lambda ^{4}-192\lambda ^{3}+144\lambda ^{2}-48\lambda +6}{(-\eta ^{2}\lambda ^{2}+\eta ^{2}\lambda -4\lambda ^{2}+4\lambda -1)^{2}}}={\frac {6-\lambda (\lambda -1)(\eta ^{4}(2\lambda (\lambda -1)-1)-48(2\lambda (\lambda -1)+1))}{(\lambda (\eta ^{2}+4)(\lambda -1)+1)^{2}}}$