We say follow 2D Normal $$ \left[\begin{array}{cc} X\ Y \end{array}\right] \sim \mathcal{N}(\mu = \left[\begin{array}{cc} \mu_{X}\ \mu_{Y}\ \end{array}\right], \Sigma = \left[\begin{array}{cc} \sigma_{X}^{2} & \sigma_{X,Y}\ \sigma_{Y,X} & \sigma_{Y}^{2} \end{array}\right] )
if they have a bivaraite normal joint pdf # Joint PDFf_{X,Y}(x,y) = \frac{1}{\sqrt{ (2\pi)^{2}
\left|\begin{array}{cc} \sigma_{X}^{2} & \sigma_{XY}\ \sigma_{YX} & \sigma_{Y}^{2}\ \end{array}\right| }}\exp { -\frac{1}{2}
\left[\begin{array}{cc} x - \mu_{X}\ y - \mu_{Y} \end{array}\right]^{T} \left[\begin{array}{cc} \sigma_{X}^{2} & \sigma_{XY}\ \sigma_{XY} & \sigma_{Y}^{2}\ \end{array}\right]^{-1} \left[\begin{array}{cc} x - \mu_{X}\ y - \mu_{Y} \end{array}\right] }, \forall x,y \in \mathbb{R}^{2}
![[Bivariate Normal-20251007135544556.webp|678]] # Properties - Marginal distributions of multivariate normal are normal - Linear transformations of normal RVs are normal - Conditional distributions of multivariate normal are normal