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${\begin{aligned}p(\mu \mid \mathbf {X} )&\propto p(\mathbf {X} \mid \mu )p(\mu )\\&=\left({\frac {\tau }{2\pi }}\right)^{n/2}\exp \left[-{\frac {1}{2}}\tau \left(\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+n({\bar {x}}-\mu )^{2}\right)\right]{\sqrt {\frac {\tau _{0}}{2\pi }}}\exp \left(-{\frac {1}{2}}\tau _{0}(\mu -\mu _{0})^{2}\right)\\&\propto \exp \left(-{\frac {1}{2}}\left(\tau \left(\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+n({\bar {x}}-\mu )^{2}\right)+\tau _{0}(\mu -\mu _{0})^{2}\right)\right)\\&\propto \exp \left(-{\frac {1}{2}}\left(n\tau ({\bar {x}}-\mu )^{2}+\tau _{0}(\mu -\mu _{0})^{2}\right)\right)\\&=\exp \left(-{\frac {1}{2}}(n\tau +\tau _{0})\left(\mu -{\dfrac {n\tau {\bar {x}}+\tau _{0}\mu _{0}}{n\tau +\tau _{0}}}\right)^{2}+{\frac {n\tau \tau _{0}}{n\tau +\tau _{0}}}({\bar {x}}-\mu _{0})^{2}\right)\\&\propto \exp \left(-{\frac {1}{2}}(n\tau +\tau _{0})\left(\mu -{\dfrac {n\tau {\bar {x}}+\tau _{0}\mu _{0}}{n\tau +\tau _{0}}}\right)^{2}\right)\end{aligned}}$

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Normal distribution - Wikipedia

${\begin{aligned}X_{k_{1},k_{2}}&=\sum _{n_{1}=0}^{N_{1}-1}\left(\sum _{n_{2}=0}^{N_{2}-1}x_{n_{1},n_{2}}\cos \left[{\frac {\pi }{N_{2}}}\left(n_{2}+{\frac {1}{2}}\right)k_{2}\right]\right)\cos \left[{\frac {\pi }{N_{1}}}\left(n_{1}+{\frac {1}{2}}\right)k_{1}\right]\\&=\sum _{n_{1}=0}^{N_{1}-1}\sum _{n_{2}=0}^{N_{2}-1}x_{n_{1},n_{2}}\cos \left[{\frac {\pi }{N_{1}}}\left(n_{1}+{\frac {1}{2}}\right)k_{1}\right]\cos \left[{\frac {\pi }{N_{2}}}\left(n_{2}+{\frac {1}{2}}\right)k_{2}\right].\end{aligned}}$

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{\displaystyle {\begin{aligned}X_{k_{1},k_{2}}&...

${\begin{aligned}A&{}=\iint _{T}\left\|\left(1,0,{\partial f \over \partial x}\right)\times \left(0,1,{\partial f \over \partial y}\right)\right\|\mathrm {d} x\,\mathrm {d} y\\&{}=\iint _{T}\left\|\left(-{\partial f \over \partial x},-{\partial f \over \partial y},1\right)\right\|\mathrm {d} x\,\mathrm {d} y\\&{}=\iint _{T}{\sqrt {\left({\partial f \over \partial x}\right)^{2}+\left({\partial f \over \partial y}\right)^{2}+1}}\,\,\mathrm {d} x\,\mathrm {d} y\end{aligned}}$

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{\displaystyle {\begin{aligned}A&{}=\iint _{T}\...

${\begin{aligned}\sigma &={\sqrt {\operatorname {E} \left((X-\mu )^{2}\right)}}\\&={\sqrt {\operatorname {E} \left(X^{2}\right)+\operatorname {E} (-2\mu X)+\operatorname {E} \left(\mu ^{2}\right)}}\\&={\sqrt {\operatorname {E} \left(X^{2}\right)-2\mu \operatorname {E} (X)+\mu ^{2}}}\\&={\sqrt {\operatorname {E} \left(X^{2}\right)-2\mu ^{2}+\mu ^{2}}}\\&={\sqrt {\operatorname {E} \left(X^{2}\right)-\mu ^{2}}}\\&={\sqrt {\operatorname {E} \left(X^{2}\right)-(\operatorname {E} (X))^{2}}}\end{aligned}}$

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표준 편차 - 위키백과, 우리 모두의 백과사전

${\begin{aligned}\left|\mu -m\right|&=\left|\mathrm {E} (X-m)\right|\\&\leqslant \mathrm {E} \left(\left|X-m\right|\right)\\&\leqslant \mathrm {E} \left(\left|X-\mu \right|\right)\\&=\mathrm {E} \left({\sqrt {(X-\mu )^{2}}}\right)\\&\leqslant {\sqrt {\mathrm {E} ((X-\mu )^{2})}}=\sigma .\end{aligned}}.$

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Odchylenie standardowe – Wikipedia, wolna encyk...

${\textstyle {\begin{aligned}1-\Phi \left(x\right)&=\left({\frac {0.39894228040143268}{x+2.92678600515804815}}\right)\left({\frac {x^{2}+8.42742300458043240x+18.38871225773938487}{x^{2}+5.81582518933527391x+8.97280659046817350}}\right)\\&\left({\frac {x^{2}+7.30756258553673541x+18.25323235347346525}{x^{2}+5.70347935898051437x+10.27157061171363079}}\right)\left({\frac {x^{2}+5.66479518878470765x+18.61193318971775795}{x^{2}+5.51862483025707963x+12.72323261907760928}}\right)\\&\left({\frac {x^{2}+4.91396098895240075x+24.14804072812762821}{x^{2}+5.26184239579604207x+16.88639562007936908}}\right)\left({\frac {x^{2}+3.83362947800146179x+11.61511226260603247}{x^{2}+4.92081346632882033x+24.12333774572479110}}\right)e^{-{\frac {x^{2}}{2}}}\end{aligned}}}$

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Normal distribution - Wikipedia

${\begin{aligned}p(\sigma ^{2}\mid \mathbf {X} )&\propto p(\mathbf {X} \mid \sigma ^{2})p(\sigma ^{2})\\&=\left({\frac {1}{2\pi \sigma ^{2}}}\right)^{n/2}\exp \left[-{\frac {S}{2\sigma ^{2}}}\right]{\frac {(\sigma _{0}^{2}{\frac {\nu _{0}}{2}})^{\frac {\nu _{0}}{2}}}{\Gamma \left({\frac {\nu _{0}}{2}}\right)}}~{\frac {\exp \left[{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right]}{(\sigma ^{2})^{1+{\frac {\nu _{0}}{2}}}}}\\&\propto \left({\frac {1}{\sigma ^{2}}}\right)^{n/2}{\frac {1}{(\sigma ^{2})^{1+{\frac {\nu _{0}}{2}}}}}\exp \left[-{\frac {S}{2\sigma ^{2}}}+{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right]\\&={\frac {1}{(\sigma ^{2})^{1+{\frac {\nu _{0}+n}{2}}}}}\exp \left[-{\frac {\nu _{0}\sigma _{0}^{2}+S}{2\sigma ^{2}}}\right]\end{aligned}}$

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Normal distribution - Wikipedia

$x_{n_{1},n_{2},n_{3}}=\sum _{k_{1}=0}^{N_{1}-1}\sum _{k_{2}=0}^{N_{2}-1}\sum _{k_{3}=0}^{N_{3}-1}X_{k_{1},k_{2},k_{3}}\cos \left[{\frac {\pi }{N_{1}}}\left(n_{1}+{\frac {1}{2}}\right)k_{1}\right]\cos \left[{\frac {\pi }{N_{2}}}\left(n_{2}+{\frac {1}{2}}\right)k_{2}\right]\cos \left[{\frac {\pi }{N_{3}}}\left(n_{3}+{\frac {1}{2}}\right)k_{3}\right],\quad {\text{for }}n_{i}=0,1,2,\dots ,N_{i}-1.$

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{\displaystyle x_{n_{1},n_{2},n_{3}}=\sum _{k_{...

$X_{k_{1},k_{2},k_{3}}=\sum _{n_{1}=0}^{N_{1}-1}\sum _{n_{2}=0}^{N_{2}-1}\sum _{n_{3}=0}^{N_{3}-1}x_{n_{1},n_{2},n_{3}}\cos \left[{\frac {\pi }{N_{1}}}\left(n_{1}+{\frac {1}{2}}\right)k_{1}\right]\cos \left[{\frac {\pi }{N_{2}}}\left(n_{2}+{\frac {1}{2}}\right)k_{2}\right]\cos \left[{\frac {\pi }{N_{3}}}\left(n_{3}+{\frac {1}{2}}\right)k_{3}\right],\quad {\text{for }}k_{i}=0,1,2,\dots ,N_{i}-1.$

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{\displaystyle X_{k_{1},k_{2},k_{3}}=\sum _{n_{...

$\left({\widehat {Tv}}\right)^{i'}={\hat {T}}_{j'}^{i'}{\hat {v}}^{j'}=\left[\left(R^{-1}\right)_{i}^{i'}T_{j}^{i}R_{j'}^{j}\right]\left[\left(R^{-1}\right)_{k}^{j'}v^{k}\right]=\left(R^{-1}\right)_{i}^{i'}(Tv)^{i}.$

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{\displaystyle \left({\widehat {Tv}}\right)^{i'...

${\begin{aligned}\mathbf {A} _{x}\!\left(\mathbf {r} ,\omega \right)&={\frac {\mu _{0}}{4\pi }}\int _{\Omega }{\frac {\mathbf {J} _{x}{\left(\mathbf {r} ',\omega \right)}}{R}}\,e^{-jkR}\,d^{3}\mathbf {r} ',\\\mathbf {A} _{y}\!\left(\mathbf {r} ,\omega \right)&={\frac {\mu _{0}}{4\pi }}\int _{\Omega }{\frac {\mathbf {J} _{y}{\left(\mathbf {r} ',\omega \right)}}{R}}\,e^{-jkR}\,d^{3}\mathbf {r} ',\\\mathbf {A} _{z}\!\left(\mathbf {r} ,\omega \right)&={\frac {\mu _{0}}{4\pi }}\int _{\Omega }{\frac {\mathbf {J} _{z}{\left(\mathbf {r} ',\omega \right)}}{R}}\,e^{-jkR}\,d^{3}\mathbf {r} '\end{aligned}}$

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{\displaystyle {\begin{aligned}\mathbf {A} _{x}...

${\begin{aligned}A_{x}{\left(\mathbf {r} ,t\right)}&={\frac {\mu _{0}}{\ 4\pi \ }}\int _{\Omega }{\frac {J_{x}{\left(\mathbf {r} ',t'\right)}}{R}}\,d^{3}\mathbf {r} '\ ,\\A_{y}{\left(\mathbf {r} ,t\right)}&={\frac {\mu _{0}}{\ 4\pi \ }}\int _{\Omega }{\frac {J_{y}{\left(\mathbf {r} ',t'\right)}}{R}}\,d^{3}\mathbf {r} '\ ,\\A_{z}{\left(\mathbf {r} ,t\right)}&={\frac {\mu _{0}}{\ 4\pi \ }}\int _{\Omega }{\frac {J_{z}{\left(\mathbf {r} ',t'\right)}}{R}}\,d^{3}\mathbf {r} '~.\end{aligned}}$

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{\displaystyle {\begin{aligned}A_{x}{\left(\mat...

$f(x,y)={\frac {1}{2\pi \sigma \tau {\sqrt {1-\rho ^{2}\ }}}}\exp \left(-{\frac {1}{2(1-\rho ^{2})}}\left(\left({x-\mu \over \sigma }\right)^{2}-2\rho \left({x-\mu \over \sigma }\right)\left({y-\nu \over \tau }\right)+\left({y-\nu \over \tau }\right)^{2}\right)\right)$