NAVIGATION

Home

Research

Bookshelf

Garden

FIND ME ON

GitHub

LinkedIn

Email

Home

Research

Bookshelf

Garden

Covariance

Definition (Covariance)

Let XX and YY be RVs, their Covariance is defined as Cov(X,Y)=E[(X−E[X])(Y−E[Y])]=E[XY]−E[X]E[Y]Cov(X,Y)=E[(X-E[X])(Y-E[Y])]=E[XY]-E[X]E[Y]

Proposition (Properties of Covariance)

  1. Linearity: For α1,α2∈R\alpha_1,\alpha_2\in\mathbb{R} Cov(X,α1Y1+α2Y2)=α1Cov(X,Y1)+α2Cov(X,Y2)Cov(X,\alpha_1Y_1+\alpha_2Y_2)=\alpha_1Cov(X,Y_1)+\alpha_2Cov(X,Y_2)
  2. Independence: If XX and YY are independent then Cov(X,Y)=0Cov(X,Y)=0

Lemma (Variance Identity)

Var(αX+βY)=α2Cov(X,X)+β2Cov(Y,Y)+2αβCov(X,Y)Var(\alpha X+\beta Y)=\alpha^2Cov(X,X)+\beta^2Cov(Y,Y)+2\alpha\beta Cov(X,Y)

Linked from

Covariance Matrix

Covariance

Multivariate Gaussian

Gaussian Process

Continuous-time Gaussian process motion planning via probabilistic inference