Concept guide
Independence of random variables: checking the complete finite joint table, with zero covariance only a consequence
(1) Joint table J₁: X∈{0,1}, Y∈{0,1,2}, P(0,0)=1/12, P(0,1)=1/12, P(0,2)=1/6, P(1,0)=1/6, P(1,1)=1/6, P(1,2)=1/3. Aggregate both marginals from the joint table, check the product equation at all 6 value pairs, and decide whether X and Y are independent; also prove that independence implies independence of any coordinate events {X∈S} and {Y∈T} and E[XY]=E[X]E[Y]. (2) Joint table J₂: X uniform on {−1,0,1} and Y=X². Prove Cov(X,Y)=0 although X and Y are not independent. (3) Use joint table J₃ to show that checking a single value pair is not enough.