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# Lecture24 - GEM2900 Understanding Uncertainty Statistical...

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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

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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[ a X + 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 ] .

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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 ] + 2Cov[ 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 ] . squaresolid If X and Y are random variables and a, b, c, d R are constants, then Cov[ aX + b, cY + d ] = a c Cov[ X, Y ] .

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The normal distribution GEM2900: Understanding Uncertainty & Statistical Thinking DSAP, NUS, Semester 1, 2008/2009 – 184 squaresolid The normal distribution, also called the Gaussian distribution, can be used to model continuous random variables.
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Lecture24 - GEM2900 Understanding Uncertainty Statistical...

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