多维正态分布

$(X, Y) \sim N (μ_{1}, μ_{2}, σ_{1}^{2}, σ_{2}^{2}, ρ)$

\begin{array}{r} f (x, y) = \frac{1}{2 π σ_{1} σ_{2} \sqrt{1 - ρ^{2}}} \exp {- \frac{1}{2 (1 - ρ^{2})} [(\frac{x - μ_{1}}{σ_{1}})^{2} - 2 ρ (\frac{x - μ_{1}}{σ_{1}}) (\frac{y - μ_{2}}{σ_{2}}) + (\frac{y - μ_{2}}{σ_{2}})^{2}]} \end{array}

使用协方差矩阵研究：
$X \sim N (μ_{1}, σ_{1}^{2}), Y \sim N (μ_{2}, σ_{2}^{2})$

\begin{aligned} C & = (\begin{array}{c} σ_{1}^{2} & ρ σ_{1} σ_{2} \\ ρ σ_{1} σ_{2} & σ_{2}^{2} \end{array}) \\ | C | & = σ_{1}^{2} σ_{2}^{2} (1 - ρ^{2}) \\ C^{- 1} & = \frac{1}{| C |} (\begin{array}{c} σ_{2}^{2} & - ρ σ_{1} σ_{2} \\ - ρ σ_{1} σ_{2} & σ_{1}^{2} \end{array}) \end{aligned}

\begin{aligned} x & = (\begin{array}{c} x_{1} \\ x_{2} \end{array}) μ = (\begin{array}{c} μ_{1} \\ μ_{2} \end{array}) \end{aligned}

\begin{aligned} f (x_{1}, x_{2}) & = \frac{1}{2 π \sqrt{| C |}} \exp [- \frac{1}{2} (x - μ)^{T} C^{- 1} (x - μ)] \end{aligned}