49 law of large numbers and central limit theorem 491

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Unformatted text preview: ther X or Y has mean zero, then E [XY ] = Cov(X, Y ). Random variables X and Y are called uncorrelated if Cov(X, Y ) = 0. (If Var(X ) > 0 and Var(Y ) > 0, so that ρX,Y is well defined, then X and Y being uncorrelated is equivalent to ρX,Y = 0.) If Cov(X, Y ) > 0, or equivalently, ρX,Y > 0, the variables are said to be positively correlated, and if Cov(X, Y ) < 0, or equivalently, ρX,Y < 0, the variables are said to be negatively correlated. If X and Y are independent, then E [XY ] = E [X ]E [Y ], which implies that X and Y are uncorrelated. The converse is false–uncorrelated does not imply independence–and in fact, independence is a much stronger condition than being uncorrelated. Specifically, independence requires a large number of equations to hold, namely FXY (u, v ) = FX (u)FY (v ) for every real value of u and v . The condition of being uncorrelated requires only a single equation to hold. Three or more random variables are said to be uncorrelated if they are...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Tech.

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