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Unformatted text preview: =0) only means no linear relationship. • It is possible for two r.v.’s to be nonlinearly correlated (and so not independent), but have zero covariance. Figure: X and Y are uncorrelated ( X 2 + Y 2 = 1 or Y = X 2 ), but they are not independent Correlation Coefficient The correlation coefficient, or just correlation, between X and Y is given by Corr ( X , Y ) = ρ XY = Cov( X , Y ) Positive covariance σ XσY = σ XY σ XσY positive correlation, and vice versa. Facts about correlation coefficient 1. ρXY lies between ‐1 and 1. 2. Variance, mean, and covariance all depend on the units variables are measured in. E.g., X=wage in $’s vs wage in $1000s 6 3. Correlation coefficient is a scale free measure of linear association. Example: If X and Y are heights in yards, and we change it to feet, then μ X , μY
will triple. σ X , σ Y will triple, Var(X) will go up 9 times, Cov(X,Y) will go up 9 times, but Corr(X,Y) will not change. Conditional Expectation*** The expected value of Y, given that we know X=x. It is denoted as E(Y|X=x). Example: Given that Y = 1.05 + .5 X + u , where u is mean zero, then E (Y | X = 12) = ? For example, suppose Y=hourly wage (in dollars) and X =years of schooling, then the conditional expectation is the expected hourly wage given years of schooling = 12(HS graduates), that is, the average hourly wage for all people with 12 years of schooling in the population. Facts about conditional expectation 1. Conditioning on X means taking X as constant, so a. For any function f(X), E[f(X)|X] = f(X). b. For any functions f(X) and g(X), we have E[f(X)Y + g(X)|X] = f(X)E(Y|X) + g(X) 2. If X and Y are independent, then E(Y|X) = E(Y). Example: Assume that IQ (denoted by u) and study time (X) are independent (which means smart people do not on average study more or less than not smart people, so information on IQ does not help update one’s expectation on study time). Also assume that the population mean of study time is 12 hours per week, then E(X|u) = E(X)=12. 3. E(Y|X) is a function of X, unless X and Y are independent, in which case E(Y|X) =E(Y), a constant. 4. Law of iterated expectations: E(E(Y|X))=E(Y) Examples: 7 1. Let Y = GPA and X = Study hours per week. S...
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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.
- Spring '08