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# Lec13 - IOE/Stat IOE/Stat 265 Fall 2009 Lecture#13#13...

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IOE/Stat 265, Fall 2009 Lecture #13: Functions of Random Variables 5-1 Jointly Distributed Random Variables 5-2 Expected Values, Covariance & Correlation 5-3 Statistics and Their Distributions 5-4 Distribution of the Sample Mean 1 5-5 Distributions of a Linear Combination 5-2 Expected Values, Covariance, nd Correlation and Correlation Definition: Expected Value of a Function of Two Random Variables – p197 () , , , ... Discrete X, Y EhXY hxy pxy ⎡⎤ = ⎣⎦ ∑∑ i , , , ... Continuous X,Y xy E h X Y h x y f x y dx dy +∞ +∞ −∞ −∞ = ∫∫ i ∞∞ 2 2 Definitions (continued) 5.2 Definitions (continued) ± Expected Value of a Function – pg 197 ± Covariance Function – pg 198, 199 ± Correlation Function – pg 200 dependence & Correlation g 200 Independence & Correlation pg 200 Scaling of Variables – pg 200 3 Example 5: Air Conditioner aintenance Maintenance S e r v i c e T i m e ± X = Service Time ± Y = Number of Units Serviced : Y: 1234 1 12.0% 8.0% 7.0% 5.0% 0% 5 0% 1 0% 3 0% X: 2 8.0% 15.0% 21.0% 13.0% 3 1.0% 1.0% 2.0% 7.0% ± uppose you pay \$8/hr + \$10/unit serviced Suppose you pay \$8/hr + \$10/unit serviced ± How much would you expect to pay ? 4

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Covariance and Correlation 5-2 Covariance and Correlation hen X and Y are ot dependent but ± When X and Y are not independent but dependent, we need to characterize the “ egree of Association” etween X and Y? Degree of Association between X and Y? he common measures are: ± The common measures are: ± Covariance (symbolically σ XY ) orrelation (symbollically ± Correlation (symbollically ρ XY ) 6 orrelation Coefficient Correlation Coefficient orrelation coefficient simply provides a scaling of ± Correlation coefficient simply provides a scaling of the covariance. ρ ,Y = Cov(X,Y) / ( σ X * σ Y ) X,Y ± If X and Y are independent then ρ = 0 ote: 0 does not imply independence ] [Note: ρ = 0 does not imply independence.] ± If Y is perfectly predictable from X (e.g. Y= aX + b) ρ = 1 implies a perfect positive linear relationship between X and Y 7 ρ = -1 implies a perfect negative linear relationship between X and Y. Example 4: Washer Quality evisited) (Revisited) ole diameter(X) and thickness(Y) varies from washer ± Hole diameter(X) and thickness(Y) varies from washer to washer according to ≤≤ + = 1 1x2 , 4y5 (x y) f(x,y) 6 otherwise 0 ± Are X and Y independent? 8 ± What is the Covariance of X and Y? 11
Example 5: Air Conditioner aintenance Maintenance ± X = Service Time ± Y = Number of Units Serviced Y: 1234 X: 1 12.0% 8.0% 7.0% 5.0% 2 8.0% 15.0% 21.0% 13.0% 3 1.0% 1.0% 2.0% 7.0% ± Are X and Y Independent ? No. (0.32)(0.21)=0.67 0.12 16 ± Calculate the Covariance σ XY and Correlation ρ XY hat is the Covariance of X and Y? ± What is the Covariance of X and Y? ± What is the Correlation between X and Y? 19 -Examp le±6±- ivariate ormal Distribution Bivariate Normal Distribution Definition 22 23

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Example 6: Bivariate Normal istribution Distribution 24 CTools Resource: Correlation.xls int Patterns for X and Y Joint Patterns for X and Y hortcut: Cov(X Y) E(XY) ± Shortcut: Cov(X,Y) = E(XY) - μ X * μ Y oncern with Cov: computed value depends ± Concern with Cov: computed value depends critically on the units of measure.
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Lec13 - IOE/Stat IOE/Stat 265 Fall 2009 Lecture#13#13...

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