More basic multivariate results

This shouldn't come as any surprise:

Cov(X,Y) = E(XY) - μ_Xμ_Y

Why not? Well, consider what happens when we let Y = X.

Cov(X,X) = E((X-μ_X)(X-μ_X)) = E((X-μ_X)2) = Var(X) = E(X²) - μ_X²

That's certainly not a proof but, given that covariance is basically an extension of variance to multivariate random variables, it is to be expected. Factoring out linear combinations also works the way you'd expect:

Var(aX+bY) = a²Var(X) + b²Var(Y) + 2abCov(X,Y)

Again, not a proof, but if you let Y=X, you get:

Var(aX+bX) = Var((a+b)X) = (a+b)²Var(X) = a²Var(X) + b²Var(X) + 2abVar(X)

Remembering what happens in the univariate case is an easy mnemonic.