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Unformatted text preview: function of X. Positive correlation (i.e. ρX,Y > 0) means that X and Y both tend to be large or both tend to be small, whereas a negative correlation (i.e. ρX,Y < 0) means that X and Y tend to be opposites: if X is larger than average it tends to indicate that Y is smaller than average. The extreme case ρX,Y = 1 means Y can be perfectly predicted by a linear function aX + b with a > 0, and the extreme case ρX,Y = −1 means Y can be perfectly predicted by a linear function aX + b with a < 0. As mentioned earlier in this section, X and Y are said to be uncorrelated if Cov(X, Y ) = 0, and being uncorrelated does not imply independence. 154 CHAPTER 4. JOINTLY DISTRIBUTED RANDOM VARIABLES Proposition 4.8.3 (Schwarz’s inequality) For two random variables X and Y : |E [XY ]| ≤ E [X 2 ]E [Y 2 ]. E [X 2 ] Furthermore, if = 0, equality holds in (4.23) (i.e. |E [XY ]| = P {Y = cX } = 1 for some constant c. (4.23) E [X 2 ]E [Y 2 ]) if and only if Proof. Take λ = E [XY ]/E [X 2 ] and note that 0 ≤ E [(Y...
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