SPLecture3 - Lecture 3 Mariana Olvera-Cravioto Columbia...

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Lecture 3 Mariana Olvera-Cravioto Columbia University [email protected] January 28th, 2015 IEOR 4106, Intro to OR: Stochastic Models Lecture 3 1/18
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Covariance and correlation I The covariance of random variables X and Y , denoted Cov( X, Y ) , is defined as Cov( X, Y ) = E [( X - E [ X ])( Y - E [ Y ])] = E [ XY ] - E [ X ] E [ Y ] I We define the correlation of X and Y as ρ ( X, Y ) = Cov( X, Y ) p Var( X )Var( Y ) I For any random variables X and Y , - 1 ρ ( X, Y ) 1 . I The correlation measures the amount of “linear” dependence between X and Y , in particular, if Y = aX , then ρ ( X, Y ) = 1 if a > 0 and ρ ( X, Y ) = - 1 if a < 0 . I We say X and Y are uncorrelated if ρ ( X, Y ) = 0 ( Cov( X, Y ) = 0 ). IEOR 4106, Intro to OR: Stochastic Models Lecture 3 2/18
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Uncorrelated is not the same as independent Uncorrelated random variables are not necessarily independent! Example: Let X be a Uniform [ - 1 , 1] random variable and set Y = X 2 , then E [ XY ] = E [ X 3 ] = Z 1 - 1 1 2 x 3 dx = 1 2 x 4 4 1 - 1 = 0 E [ X ] = Z 1 - 1 1 2 x dx = 1 2 x 2 2 1 - 1 = 0 E [ Y ] = E [ X 2 ] = Z 1 - 1 1 2 x 2 dx = 1 2 x 3 3 1 - 1 = 1 3 therefore, Cov( X, Y ) = E [ XY ] - E [ X ] E [ Y ] = 0 IEOR 4106, Intro to OR: Stochastic Models Lecture 3 3/18
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Long example Suppose that X and Y have the joint PDF f ( x, y ) = ( 24 xy, for x 0 , y 0 , x + y 1 0 , otherwise. Compute Cov( X, Y ) . IEOR 4106, Intro to OR: Stochastic Models Lecture 3 4/18
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Long example Suppose that X and Y have the joint PDF f ( x, y ) = ( 24 xy, for x 0 , y 0 , x + y 1 0 , otherwise. Compute Cov( X, Y ) . We start by computing the marginal PDFs: f X ( x ) = Z 1 - x 0 24 xy dy = 12 xy 2 1 - x 0 = 12 x (1 - x ) 2 , for 0 x 1 f Y ( y ) = Z 1 - y 0 24 xy dx = 12 x 2 y 1 - y 0 = 12 y (1 - y ) 2 , for 0 y 1 IEOR 4106, Intro to OR: Stochastic Models Lecture 3 4/18
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Long example continued We now compute the expectations E [ X ] , E [ Y ] and E [ XY ] . IEOR 4106, Intro to OR: Stochastic Models Lecture 3 5/18
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Long example continued We now compute the expectations E [ X ] , E [ Y ] and E [ XY ] . E [ X ] = Z 1 0 xf X ( x ) dx = Z 1 0 12 x 2 (1 - x ) 2 dx = 12 x 3 3 - 2 x 4 4 + x 5 5 1 0 = 2 5 E [ Y ] = Z 1 0 yf Y ( y ) dy = Z 1 0 12 y 2 (1 - y ) 2 dy = 12 y 3 3 - 2 y 4 4 + y 5 5 1 0 = 2 5 E [ XY ] = Z 1 0 Z 1 - x 0 xyf ( x, y ) dy dx = Z 1 0 x 2 Z 1 - x 0 24 y 2 dy dx = Z 1 0 8 x 2 (1 - x ) 3 dx = 8 x 3 3 - 3 x 4 4 + 3 x 5 5 - x 6 6 1 0 = 2 15 Therefore, Cov( X, Y ) = E [ XY ] - E [ X ] E [ Y ] = 2 15 - 2 5 · 2 5 = - 2 75 IEOR 4106, Intro to OR: Stochastic Models Lecture 3 5/18
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Long example continued We can also compute E [ X 2 ] and E [ Y 2 ] to compute the variances of X and Y and use them to obtain the correlation. IEOR 4106, Intro to OR: Stochastic Models Lecture 3 6/18
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Long example continued We can also compute E [ X 2 ] and E [ Y 2 ] to compute the variances of X and Y and use them to obtain the correlation. E [ X 2 ] = E [ Y 2 ] = Z 1 0 y 2 f Y ( y ) dy = Z 1 0 12 y 3 (1 - y ) 2 dy = 12 y 4 4 - 2 y 5 5 + y 6 6 1 0 = 1 5 so Var( X ) = Var( Y ) = E [ Y 2 ] - ( E [ Y ]) 2 = 1 5 - 2 5 2 = 1 25 and ρ ( X, Y ) = Cov( X, Y ) p Var( X )Var( Y ) = - 2 75 1 25 = - 2 3 IEOR 4106, Intro to OR: Stochastic Models Lecture 3 6/18
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Markov inequality I In practice, we hardly ever now the entire distribution of a r.v. I It can be very useful to at least be able to bound probabilities.
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