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Unformatted text preview: Var and sd of a constant is 0. 4. For any constants a and b, Var ( X + b) = Var ( X ) , Var (aX ) = a 2Var ( X ) , Var (aX + b) = a 2Var ( X ) , and sd ( aX + b) =| a | sd ( X ) • Adding a constant to a random variable does not change the variance or sd! Adding a constant to a r. v. = shift the location of the whole distribution, and so would not change the variance or more generally the shape of the distribution. 4 Figure Example: demeaning a r.v. X, which means subtracting off the mean from X so that it is mean zero, the variance of X does not change. Var ( X − μ X ) = Var ( X ) 5. For any constants a and b, Var (aX + bY ) = a 2Var ( X ) + b2Var (Y ) + 2abCov( X , Y ) Var ( X − Y ) = ? Var ( X + Y ) = ? Covariance of two r.v.’s For r.v.’s X and Y Cov( X , Y ) = E[( X − μ X )(Y − μY )] = E[( X − μ X )Y ] = E[ X (Y − μY )] = E ( XY ) − μ X μY Want to know on average how two r.v.s vary with each other. Example: Health insurance and health Assume we have the following possible outcomes and probabilities Selfreported health(Y) 3(good/excellent)
3 2(fair) 2 1(bad) 1 Health Insurance(X) 1 0 1 0 1 0 Prob. 0.17 0.08 0.21 0.14 0.12 0.28 E(Y)=2; E(X)=0.5 Cov(X, Y)=0.17(3‐2)(1‐0.5)+0.08(3‐2)(0‐0.5)+0.21(2‐2)(1‐0.5)+0.14(2‐2)(0‐
0.5)+0.12(1‐2)(1‐0.5)+0.28(1‐2)(0‐0.5)=0.05 5 So, health insurance and self‐reported are positively correlated; those who have health insurance tend to report better health. Depending on the coding of self‐reported health, the absolute value of the variance may change. Facts about covariance 1. Covariance is a measure of linear association between two r.v.’s. o For two r.v.s u and X, if E(u|X)=0, i.e., u is conditional mean zero, conditional on X, then u and X are uncorrelated. o The converse is not true. E.g., if u = X 2 , then they are uncorrelated (why?), but E (u | X ) = X 2 ≠ 0 . 2. If X and u are independent, then Cov ( X , u ) = 0 , i.e., X and u are not correlated. The converse is not true! • Independence means lack of any relationships (both linear and nonlinear), while being not correlated or zero correlation (covariance...
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This note was uploaded on 02/17/2014 for the course PSY 2000 taught by Professor Staff during the Spring '08 term at University of Texas at Austin.
- Spring '08