高斯分布简述

A Brief of Gaussian Distribution
Gaussian distribution, also known as normal distribution, is the most popular continuous probability distribution.

In monovariate case, x∈(−∞,∞), parameters of the Gaussian distribution are average μ∈(−∞,∞)and variance σ2>0. The probability function is defined as follows

p(x|μ,σ2)=12πσ2−−−−√exp{−(x−μ)22σ2}

E[x]=μ

var[x]=σ2

In multivariate case, consider the d-dimension vector x, the average μ is a d-dimension vector, too. However, the covariance in this case becomes a positive definite matrix Σ of d×d dimension.

p(x|μ,Σ)=1(2π)ddet(Σ)−−−−−−−−−−√exp{−12(x−μ)TΣ−1(x−μ)}

E[x]=μ

cov[x]=Σ