2次元正規分布

${\displaystyle p\left(x_1,x_2\right)=\frac{1}{2\pi \sqrt{{\sigma }_x^2{\sigma }_y^2\left(1-{\rho }^2\right)}}{e}^{-\frac{1}{2\left(1-{\rho }^2\right)}\left(\frac{{\left(x_1-{\mu }_1\right)}^2}{{\sigma }_1^2}-\frac{2\rho \left(x_1-{\mu }_1\right)\left(x_2-{\mu }_2\right)}{{\sigma }_1{\sigma }_2}+\frac{{\left(x_2-{\mu }_2\right)}^2}{{\sigma }_2^2}\right)}}$   ･･････(1)

ただし，${\displaystyle {\mu }_1=E\left(X_1\right)}$  ( ${\displaystyle X_1}$ の平均)  ･･･(2)， ${\displaystyle {\mu }_2=E\left(X_2\right)}$  ( ${\displaystyle X_2}$ の平均)  ･･･(3)，
${\displaystyle {\sigma }_1=\sqrt{V\left(X_1\right)}}$   ( ${\displaystyle X_1}$ の標準偏差)   ･･･(4) ， ${\displaystyle {\sigma }_2=\sqrt{V\left(X_2\right)}}$  ( ${\displaystyle X_2}$ の標準偏差)   ･･･(5) ，
${\displaystyle {\sigma }_{12}=C\left(X_1,X_2\right)}$  (${\displaystyle X_1}$， ${\displaystyle X_2}$ の共分散)   ･･･(6) ， ${\displaystyle \rho =\frac{{\sigma }_{xy}}{{\sigma }_x{\sigma }_y}}$  (${\displaystyle X_1}$， ${\displaystyle X_2}$ の相関係数)   ･･･(7)