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Definition: cov(X, Y ) = E(X − µX)(Y − µY ). This can be simplified as follows:.

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Download Covariance and correlation Let random variables X, Y with ... and more Schemes and Mind Maps Economic statistics in PDF only on Docsity! Covariance and correlation Let random variables X, Y with means µX , µY respectively. The covariance, denoted with cov(X,Y ), is a measure of the association between X and Y . Definition: cov(X,Y ) = E(X − µX)(Y − µY ) This can be simplified as follows: cov(X,Y ) = E(X − µX)(Y − µY ) = E(XY )− µY E(X)− µXE(Y ) + µXµY Therefore, cov(X,Y ) = E(XY )− (EX)(EY ) Note: If X, Y are independent then E(XY ) = (EX)E(Y ) Therefore cov(X,Y ) = 0. Let W,X, Y, Z random variables, and a, b, c, d constants: • Find cov(a+X,Y ) cov(a+X,Y ) = E(a+X − µa+X)(Y − µY ) = E(a+X − µX − a)(Y − µY ) Therefore, cov(a+X,Y ) = cov(X,Y ). • Find cov(aX, bY ) cov(aX, bY ) = E(aX − µaX)(bY − µbY ) = E(aX − aµX)(bY − bµY ) Therefore, cov(aX, bY ) = abE(X − µX)(Y − µY ) = ab cov(X,Y ) • Find cov(X,Y + Z) cov(X,Y + Z) = E(X − µX)(Y − µY +Z) = E(X − µX)(Y + Z − µY − µZ) Or cov(X,Y + Z) = E(X − µX)(Y − µY + Z − µZ) = E(X − µX)(Y − µY ) + E(X − µX)(Z − µZ) Therefore, cov(X,Y + Z) = cov(X,Y ) + cov(X,Z) • Using the results above we can find cov(aW + bX, cY + dZ). cov(aW + bX, cY + dZ) = ab cov(W,Y ) + ad cov(W,Z) + bc cov(X,Y ) + bd cov(X,Z) • Correlation: However, the covariance depends on the scale of measurement and so it is not easy to say whether a particular covariance is small or large. The problem is solved by standardize the value of covariance (divide it by σXσY ), to get the so called coefficient of correlation ρXY . ρ = cov(X,Y ) σXσY , Always, −1 ≤ ρ ≤ 1 cov(X,Y ) = ρσXσY If X,Y are independent then ρXY = 0 1 • Important: var(X + Y ) = var(X) + var(Y ) + 2cov(X,Y ) Proof: • Find var(aX + bY ) • In general: Let X1, X2, · · · , Xn be random variables, and a1, a2, · · · , an be constants. Find the variance of the linear combination Y = a1X1 + a2X2 + · · ·+ anXn. 2