Concept guide
Covariance: symmetry, the affine rule, and why zero covariance does not imply independence
On one exact finite joint law: (1) prove Cov(Y,X)=Cov(X,Y); (2) prove Cov(aX+b,cY+d)=ac·Cov(X,Y); (3) prove that independence of X and Y gives Cov(X,Y)=0, so a nonzero covariance refutes independence; (4) for the joint law with X uniform on −1, 0, 1 and Y=X², compute E[X], E[Y], E[XY] and Cov(X,Y)=0, and prove that X and Y are not independent.