在概率论中，二元正态分布是一类常见的连续型二元概率分布，它可以由正态分布导出，是多元正态分布的二元特例。
设有二维连续型随机向量 X = ( X , Y ) {\displaystyle \boldsymbol{X} = (X, Y)} ，如果它的联合概率密度函数为 f ( x , y ) = 1 2 π σ 1 σ 2 1 − ρ 2 exp { − 1 2 ( 1 − ρ 2 ) [ ( x − μ 1 ) 2 σ 1 2 − 2 ρ ( x − μ 1 ) ( y − μ 2 ) σ 1 σ 2 + ( y − μ 2 ) 2 σ 2 2 ] } . {\displaystyle f(x, y) = \dfrac{1}{2\pi \sigma_1 \sigma_2 \sqrt{1-\rho^2}} \exp \left\{ \dfrac{-1}{2(1-\rho^2)} \left[ \dfrac{(x-\mu_1)^2}{\sigma_1^2} - 2\rho\dfrac{(x-\mu_1)(y-\mu_2)}{\sigma_1 \sigma_2} + \dfrac{(y-\mu_2)^2}{\sigma_2^2} \right] \right\}.} 其中， μ 1 , μ 2 ∈ R ; σ 1 , σ 2 > 0 ; − 1 < ρ < 1 {\displaystyle \mu_1, \mu_2 \in \R; \sigma_1, \sigma_2 > 0; -1 < \rho < 1} 为常数，我们就称随机向量 X {\displaystyle \boldsymbol{X}} 服从二元正态分布，记作 X ∼ N ( μ 1 , μ 2 , σ 1 2 , σ 2 2 , ρ ) . {\displaystyle \boldsymbol{X} \sim N(\mu_1, \mu_2, \sigma_1^2, \sigma_2^2, \rho).}
它是 n {\displaystyle n} 元正态分布当 n = 2 {\displaystyle n=2} 的情形，对应的 μ = ( μ 1 , μ 2 ) , Σ = ( σ 1 2 ρ σ 1 σ 2 ρ σ 1 σ 2 σ 2 2 ) {\displaystyle \boldsymbol{\mu} = (\mu_1, \mu_2), \qquad \mathbf{\Sigma} = \begin{pmatrix} \sigma_1^2 & \rho\sigma_1\sigma_2 \\ \rho\sigma_1\sigma_2 & \sigma_2^2 \end{pmatrix}}
设有上式定义的二元正态分布，那么它关于 x , y {\displaystyle x, y} 的两个边际分布的概率密度函数分别是 f X ( x ) = ∫ − ∞ + ∞ f ( x , y ) d y = 1 2 π σ 1 exp [ − ( x − μ 1 ) 2 2 σ 1 2 ] , f Y ( y ) = ∫ − ∞ + ∞ f ( x , y ) d x = 1 2 π σ 2 exp [ − ( y − μ 2 ) 2 2 σ 2 2 ] . {\displaystyle \begin{align} f_X (x) & = \int_{-\infty}^{+\infty} f(x, y) \mathrm{d}y = \dfrac{1}{\sqrt{2\pi}\sigma_1} \exp \left[ - \dfrac{(x-\mu_1)^2}{2\sigma_1^2} \right], \\ f_Y (y) & = \int_{-\infty}^{+\infty} f(x, y) \mathrm{d}x = \dfrac{1}{\sqrt{2\pi}\sigma_2} \exp \left[ - \dfrac{(y-\mu_2)^2}{2\sigma_2^2} \right]. \end{align}} 即 X ∼ N ( μ 1 , σ 1 2 ) , Y ∼ N ( μ 2 , σ 2 2 ) . {\displaystyle X \sim N(\mu_1, \sigma_1^2), Y \sim N(\mu_2, \sigma_2^2).} ，当且仅当 ρ = 0 {\displaystyle \rho = 0} 的时候有 f ( x , y ) = f X ( x ) f Y ( y ) {\displaystyle f(x, y) = f_X(x) f_Y (y)} ，即随机变量 X , Y {\displaystyle X, Y} 是相互独立的。
设有上式定义的二元正态分布，那么它的两个条件分布 f ( x | y ) , f ( y | x ) {\displaystyle f(x|y), f(y|x)} 的概率密度函数分别是 f ( x | y ) = f ( x , y ) f Y ( y ) = 1 2 π σ 1 1 − ρ 2 exp { − [ x − ( μ 1 + ρ σ 1 σ 2 ( y − μ 2 ) ) ] 2 2 σ 1 2 ( 1 − ρ 2 ) } , f ( y | x ) = f ( x , y ) f X ( x ) = 1 2 π σ 2 1 − ρ 2 exp { − [ y − ( μ 2 + ρ σ 2 σ 1 ( x − μ 1 ) ) ] 2 2 σ 2 2 ( 1 − ρ 2 ) } . {\displaystyle \begin{align} f(x|y) & = \dfrac{f(x, y)}{f_Y (y)} = \dfrac{1}{\sqrt{2\pi}\sigma_1\sqrt{1-\rho^2}} \exp \left\{ - \dfrac{ \left[ x - \left( \mu_1 + \rho \frac{\sigma_1}{\sigma_2} (y - \mu_2) \right) \right]^2}{2\sigma_1^2(1-\rho^2)} \right\}, \\ f(y|x) & = \dfrac{f(x, y)}{f_X (x)} = \dfrac{1}{\sqrt{2\pi}\sigma_2\sqrt{1-\rho^2}} \exp \left\{ - \dfrac{ \left[ y - \left( \mu_2 + \rho \frac{\sigma_2}{\sigma_1} (x - \mu_1) \right) \right]^2}{2\sigma_2^2(1-\rho^2)} \right\}. \end{align}} 它们都是正态分布 X | Y ∼ N ( μ 1 + ρ σ 1 σ 2 ( y − μ 2 ) , σ 1 2 ( 1 − ρ 2 ) ) , Y | X ∼ N ( μ 2 + ρ σ 2 σ 1 ( x − μ 1 ) , σ 2 2 ( 1 − ρ 2 ) ) . {\displaystyle X|Y \sim N \left( \mu_1 + \rho \frac{\sigma_1}{\sigma_2} (y - \mu_2), \sigma_1^2(1-\rho^2) \right), Y|X \sim N \left( \mu_2 + \rho \frac{\sigma_2}{\sigma_1} (x - \mu_1), \sigma_2^2(1-\rho^2) \right).}
978-7-0402-8890-2
，如果它的联合概率密度函数为
![{\displaystyle f(x,y)={\dfrac {1}{2\pi \sigma _{1}\sigma _{2}{\sqrt {1-\rho ^{2}}}}}\exp \left\{{\dfrac {-1}{2(1-\rho ^{2})}}\left[{\dfrac {(x-\mu _{1})^{2}}{\sigma _{1}^{2}}}-2\rho {\dfrac {(x-\mu _{1})(y-\mu _{2})}{\sigma _{1}\sigma _{2}}}+{\dfrac {(y-\mu _{2})^{2}}{\sigma _{2}^{2}}}\right]\right\}.}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/2c94123a10031ec5bdf08ce5c24fe437c40d2fd0)
其中，
为常数，我们就称随机向量
服从二元正态分布，记作
元正态分布当
的情形，对应的
的两个边际分布的概率密度函数分别是
![{\displaystyle {\begin{aligned}f_{X}(x)&=\int _{-\infty }^{+\infty }f(x,y)\mathrm {d} y={\dfrac {1}{{\sqrt {2\pi }}\sigma _{1}}}\exp \left[-{\dfrac {(x-\mu _{1})^{2}}{2\sigma _{1}^{2}}}\right],\\f_{Y}(y)&=\int _{-\infty }^{+\infty }f(x,y)\mathrm {d} x={\dfrac {1}{{\sqrt {2\pi }}\sigma _{2}}}\exp \left[-{\dfrac {(y-\mu _{2})^{2}}{2\sigma _{2}^{2}}}\right].\end{aligned}}}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/53d0bdbcec8e3b44b014fd1daf95c2c28180b16f)
即
，当且仅当
的时候有
，即随机变量
是相互独立的。
的概率密度函数分别是
![{\displaystyle {\begin{aligned}f(x|y)&={\dfrac {f(x,y)}{f_{Y}(y)}}={\dfrac {1}{{\sqrt {2\pi }}\sigma _{1}{\sqrt {1-\rho ^{2}}}}}\exp \left\{-{\dfrac {\left[x-\left(\mu _{1}+\rho {\frac {\sigma _{1}}{\sigma _{2}}}(y-\mu _{2})\right)\right]^{2}}{2\sigma _{1}^{2}(1-\rho ^{2})}}\right\},\\f(y|x)&={\dfrac {f(x,y)}{f_{X}(x)}}={\dfrac {1}{{\sqrt {2\pi }}\sigma _{2}{\sqrt {1-\rho ^{2}}}}}\exp \left\{-{\dfrac {\left[y-\left(\mu _{2}+\rho {\frac {\sigma _{2}}{\sigma _{1}}}(x-\mu _{1})\right)\right]^{2}}{2\sigma _{2}^{2}(1-\rho ^{2})}}\right\}.\end{aligned}}}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/be2251a156fcb3af9e9d9458b02a500372e93561)
它们都是正态分布
