A self-improvement to the Cauchy–Schwarz inequality

We present a self improvement to the Cauchy–Schwarz inequality, which in the probability case yields [ E ( X Y ) ] 2 ≤ E ( X 2 ) E ( Y 2 ) − ( | E ( X ) | Var ( Y ) − | E ( Y ) | Var ( X ) ) 2 . It is to be noted that the additional term to the inequality only involves the marginal first two moments for X and Y , and not any joint property. We also provide the discrete improvement to the inequality.