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${\begin{aligned}\mu _{X\cup Y}&={\frac {1}{N_{X\cup Y}}}\left(N_{X}\mu _{X}+N_{Y}\mu _{Y}-N_{X\cap Y}\mu _{X\cap Y}\right)\\[3pt]\sigma _{X\cup Y}&={\sqrt {\frac {[N_{X}-1]\sigma _{X}^{2}+N_{X}\mu _{X}^{2}+[N_{Y}-1]\sigma _{Y}^{2}+N_{Y}\mu _{Y}^{2}-[N_{X\cap Y}-1]\sigma _{X\cap Y}^{2}-N_{X\cap Y}\mu _{X\cap Y}^{2}-[N_{X}+N_{Y}-N_{X\cap Y}]\mu _{X\cup Y}^{2}}{N_{X\cup Y}-1}}}\end{aligned}}$

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Τυπική απόκλιση - Βικιπαίδεια

${\begin{aligned}\mu _{X\cup Y}&={\frac {1}{N_{X\cup Y}}}\left(N_{X}\mu _{X}+N_{Y}\mu _{Y}-N_{X\cap Y}\mu _{X\cap Y}\right)\\\sigma _{X\cup Y}&=\scriptstyle {\sqrt {{\frac {1}{N_{X\cup Y}-1}}\left([N_{X}-1]\sigma _{X}^{2}+N_{X}\mu _{X}^{2}+[N_{Y}-1]\sigma _{Y}^{2}+N_{Y}\mu _{Y}^{2}-[N_{X\cap Y}-1]\sigma _{X\cap Y}^{2}-N_{X\cap Y}\mu _{X\cap Y}^{2}-[N_{X}+N_{Y}-N_{X\cap Y}]\mu _{X\cup Y}^{2}\right)}}\end{aligned}}$

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Desviació tipus - Viquipèdia, l'enciclopèdia ll...

${\begin{aligned}\mu _{X\cup Y}&={\frac {1}{N_{X\cup Y}}}\left(N_{X}\mu _{X}+N_{Y}\mu _{Y}-N_{X\cap Y}\mu _{X\cap Y}\right)\\[3pt]\sigma _{X\cup Y}&={\sqrt {{\frac {1}{N_{X\cup Y}}}\left(N_{X}[\sigma _{X}^{2}+\mu _{X}^{2}]+N_{Y}[\sigma _{Y}^{2}+\mu _{Y}^{2}]-N_{X\cap Y}[\sigma _{X\cap Y}^{2}+\mu _{X\cap Y}^{2}]\right)-\mu _{X\cup Y}^{2}}}\end{aligned}}$

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Τυπική απόκλιση - Βικιπαίδεια

${\begin{aligned}\mu _{X\cup Y}&={\frac {1}{N_{X\cup Y}}}\left(N_{X}\mu _{X}+N_{Y}\mu _{Y}-N_{X\cap Y}\mu _{X\cap Y}\right)\\\sigma _{X\cup Y}&={\sqrt {{\frac {1}{N_{X\cup Y}}}\left(N_{X}[\sigma _{X}^{2}+\mu _{X}^{2}]+N_{Y}[\sigma _{Y}^{2}+\mu _{Y}^{2}]-N_{X\cap Y}[\sigma _{X\cap Y}^{2}+\mu _{X\cap Y}^{2}]\right)-\mu _{X\cup Y}^{2}}}\end{aligned}}$

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Desviació tipus - Viquipèdia, l'enciclopèdia ll...

$\sigma _{X\cup Y}={\sqrt {\frac {(N_{X}-1)\sigma _{X}^{2}+N_{X}\mu _{X}^{2}+(N_{Y}-1)\sigma _{Y}^{2}+N_{Y}\mu _{Y}^{2}-(N_{X\cap Y}-1)\sigma _{X\cap Y}^{2}-N_{X\cap Y}\mu _{X\cap Y}^{2}-(N_{X}+N_{Y}-N_{X\cap Y})\mu _{X\cup Y}^{2}}{N_{X}+N_{Y}-N_{X\cap Y}-1}}}\,\!$

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मानक विचलन - विकिपीडिया

${\begin{aligned}\mu _{X\cup Y}&={\frac {1}{N_{X\cup Y}}}\left(N_{X}\mu _{X}+N_{Y}\mu _{Y}\right)\\[3pt]\sigma _{X\cup Y}&={\sqrt {{\frac {1}{N_{X\cup Y}-1}}\left([N_{X}-1]\sigma _{X}^{2}+N_{X}\mu _{X}^{2}+[N_{Y}-1]\sigma _{Y}^{2}+N_{Y}\mu _{Y}^{2}-[N_{X}+N_{Y}]\mu _{X\cup Y}^{2}\right)}}\end{aligned}}$

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Τυπική απόκλιση - Βικιπαίδεια

${\begin{aligned}\mu _{X\cup Y}&={\frac {1}{N_{X\cup Y}}}\left(N_{X}\mu _{X}+N_{Y}\mu _{Y}\right)\\\sigma _{X\cup Y}&={\sqrt {{\frac {1}{N_{X\cup Y}-1}}\left([N_{X}-1]\sigma _{X}^{2}+N_{X}\mu _{X}^{2}+[N_{Y}-1]\sigma _{Y}^{2}+N_{Y}\mu _{Y}^{2}-[N_{X}+N_{Y}]\mu _{X\cup Y}^{2}\right)}}\end{aligned}}$

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Desviació tipus - Viquipèdia, l'enciclopèdia ll...

${\begin{aligned}\mu _{X\cup Y}&={\frac {N_{X}\mu _{X}+N_{Y}\mu _{Y}}{N_{X}+N_{Y}}}\\[3pt]\sigma _{X\cup Y}&={\sqrt {{\frac {N_{X}\sigma _{X}^{2}+N_{Y}\sigma _{Y}^{2}}{N_{X}+N_{Y}}}+{\frac {N_{X}N_{Y}}{(N_{X}+N_{Y})^{2}}}(\mu _{X}-\mu _{Y})^{2}}}\end{aligned}}$

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Τυπική απόκλιση - Βικιπαίδεια

${\begin{aligned}\mu _{X\cup Y}&={\frac {N_{X}\mu _{X}+N_{Y}\mu _{Y}}{N_{X}+N_{Y}}}\\\sigma _{X\cup Y}&={\sqrt {{\frac {N_{X}\sigma _{X}^{2}+N_{Y}\sigma _{Y}^{2}}{N_{X}+N_{Y}}}+{\frac {N_{X}N_{Y}}{(N_{X}+N_{Y})^{2}}}(\mu _{X}-\mu _{Y})^{2}}}\end{aligned}}$

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Desviació tipus - Viquipèdia, l'enciclopèdia ll...

$\sigma _{X\cup Y}={\sqrt {{\frac {N_{X}(\sigma _{X}^{2}+\mu _{X}^{2})+N_{Y}(\sigma _{Y}^{2}+\mu _{Y}^{2})-N_{X\cap Y}(\sigma _{X\cap Y}^{2}+\mu _{X\cap Y}^{2})}{N_{X}+N_{Y}-N_{X\cap Y}}}-\mu _{X\cup Y}^{2}}}\,\!$

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मानक विचलन - विकिपीडिया

${\begin{aligned}E(X)&={\frac {1}{{\sqrt {2}}{\sqrt {\pi }}\sigma }}\int _{-\infty }^{+\infty }(y{\sqrt {2}}\sigma +\mu )e^{-y^{2}}{\sqrt {2}}\sigma dy\\&={\frac {1}{\sqrt {\pi }}}\int _{-\infty }^{+\infty }(y{\sqrt {2}}\sigma +\mu )e^{-y^{2}}dy\\&={\frac {1}{\sqrt {\pi }}}{\Bigg (}{\sqrt {2}}\sigma \int _{-\infty }^{+\infty }ye^{-y^{2}}dy+\mu \int _{-\infty }^{+\infty }e^{-y^{2}}dy{\Bigg )}\\&={\frac {{\sqrt {2}}\sigma }{\sqrt {\pi }}}{\Bigg (}\int _{-\infty }^{+\infty }ye^{-y^{2}}dy{\Bigg )}_{=0}^{(1)}+{\frac {\mu }{\sqrt {\pi }}}{\Bigg (}\int _{-\infty }^{+\infty }e^{-y^{2}}dy{\Bigg )}_{={\sqrt {\pi }}}^{(2)}\\&={\frac {{\sqrt {2}}\sigma }{\sqrt {\pi }}}\cdot 0+{\frac {\mu }{\sqrt {\pi }}}\cdot {\sqrt {\pi }}\\&=\mu \end{aligned}}$

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Rozkład normalny – Wikipedia, wolna encyklopedia

$\sigma _{X\cup Y}={\sqrt {\frac {(N_{X}-1)\sigma _{X}^{2}+N_{X}\mu _{X}^{2}+(N_{Y}-1)\sigma _{Y}^{2}+N_{Y}\mu _{Y}^{2}-(N_{X}+N_{Y})\mu _{X\cup Y}^{2}}{N_{X}+N_{Y}-1}}}\,\!$

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मानक विचलन - विकिपीडिया

P(x_1<=x<=x_2,y_1<=y<=y_2)=D(x_1,y_1)-D(x_1,y_2)-D(x_2,y_1)+D(x_2,y_2).

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Distribution Function -- from Wolfram MathWorld

$\mu _{X\cup Y}={\frac {N_{X}\mu _{X}+N_{Y}\mu _{Y}-N_{X\cap Y}\mu _{X\cap Y}}{N_{X}+N_{Y}-N_{X\cap Y}}}\,\!$

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मानक विचलन - विकिपीडिया

${\begin{aligned}&&N_{X\cup Y}&=N_{X}+N_{Y}-N_{X\cap Y}\\X\cap Y=\varnothing &\Rightarrow &N_{X\cap Y}&=0\\&\Rightarrow &N_{X\cup Y}&=N_{X}+N_{Y}\end{aligned}}$

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Desviació tipus - Viquipèdia, l'enciclopèdia ll...

${\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}\cdot {\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\end{pmatrix}}=x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}\,.$

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{\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\\x_...

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

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