Hello,

The covariance is a bilinear and symmetric form.

So Cov(X,Y+Z)=Cov(X,Y)+Cov(X,Z)

Many of the properties of covariance can be extracted elegantly by observing that it satisfies similar properties to those of an inner product:

(1) bilinear: for constants a and b and random variables X, Y, and U, Cov(aX + bY, U) = a Cov(X, U) + b Cov(Y, U)

(2) symmetric: Cov(X, Y) = Cov(Y, X)

(3) positive semi-definite: Var(X) = Cov(X, X) ≥ 0, and Cov(X, X) = 0 implies that X is a constant random variable (K).