Normal Random Vectors

Key Concepts

• Joint Normality
• Joint Normality and Independence
• Linear Transform of Joint Normals
• Complex Joint Normals

Jointly Gaussian

The PDF of a jointly Gaussian is f_{\bf X}({\bf x}) = \frac{1}{ \sqrt{(2\pi)^N \textrm{det}\left({\bf C}_{\bf XX}\right) } } \exp\left\{ - \frac{1}{2} \left({\bf x-m_X}\right)^{T} \,{\bf C}_{\bf XX}^{-1} \, \left({\bf x-m_X}\right) \right\}. \ The exponent is a quadractic form with positive definite {\bf C}_{\bf XX}^{-1}, i.e., the surface has a bowl shape.

Non-jointly Gaussians

For jointly Gaussians, any linear projection must be Gaussian: Y = {\bf a}^H {\bf X} Visually spoken, the projection from every angle must be Gaussian. The marginals are only two specific projections.