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Unformatted text preview: GEM2900: Understanding Uncertainty & Statistical Thinking DSAP, NUS, Semester 1, 2008/2009 – 1 GEM2900: Understanding Uncertainty & Statistical Thinking Berwin Turlach [email protected] Department of Statistics and Applied Probability National University of Singapore Expectation, Covariance and Correlation (cont.) GEM2900: Understanding Uncertainty & Statistical Thinking DSAP, NUS, Semester 1, 2008/2009 – 180 The correlation coefficient of X and Y , denoted by Corr[ X,Y ] , ρ X,Y or just ρ , is defined by ρ X,Y = Cov[ X,Y ] σ X σ Y It can be shown that 1 ≤ ρ X,Y ≤ 1 . Two random variables X and Y are called uncorrelated if ρ X,Y = 0 , i.e. if Cov[ X,Y ] = 0 . Rules for Expected Values, Variances and Covariances GEM2900: Understanding Uncertainty & Statistical Thinking DSAP, NUS, Semester 1, 2008/2009 – 181 squaresolid If X is a random variable and a,b ∈ R are constants, then E[ aX + b ] = a E[ X ] + b . squaresolid If X and Y are random variables, then E[ X + Y ] = E[ X ] + E[ Y ] . squaresolid For any random variable X we have Var[ X ] = E[ X 2 ] E[ X ] 2 = E[ X 2 ] μ 2 X . squaresolid If X is a random variable and a,b ∈ R are constants, then Var[ aX + b ] = a 2 Var[ X ] . In particular, SD[ aX + b ] =  a  SD[ X ] . Rules for Expected Values, Variances and Covariances (cont.) GEM2900: Understanding Uncertainty & Statistical Thinking DSAP, NUS, Semester 1, 2008/2009 – 182 squaresolid If X and Y are independent, then E[ X Y ] = E[ X ] E[ Y ] . squaresolid If X and Y are independent then Cov[ X,Y ] = 0 , i.e. the two random variables are uncorrelated. squaresolid Var[ X + Y ] = Var[ X ] + Var[ Y ] + 2 Cov[ X,Y ] . squaresolid If X and Y are uncorrelated, then Var[ X + Y ] = Var[ X ] + Var[ Y ] . Rules for Expected Values, Variances and Covariances (cont.) GEM2900: Understanding Uncertainty & Statistical Thinking DSAP, NUS, Semester 1, 2008/2009 – 183 squaresolid For any random variable X we have Var[ X ] = Cov[ X,X ] ....
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This note was uploaded on 03/19/2012 for the course SCIENCE 2900 taught by Professor Forgot during the Spring '08 term at National University of Singapore.
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