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# JMF-6-07 - Bivariate Sampling Distributions Chapters 5.10 6...

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CH Reilly UCF - IEMS - STA 3032 1 Bivariate, Sampling Distributions Chapters 5.10 & 6 Note Set 6 Spring 2007

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CH Reilly UCF - IEMS - STA 3032 2 Bivariate random variables In many practical situations, we may be interested in analyzing or modeling random vectors, that is, random variables with more than one random component. We will only consider bivariate random variables, that is, those with 2 random components. Understand that what we discuss for bivariate random variables may be extended to multivariate random variables.
CH Reilly UCF - IEMS - STA 3032 3 Bivariate random variables (cont’d.) = = = = = 1 2 all 2 1 2 2 all 2 1 1 1 2 2 1 1 2 1 ) , ( ) ( ) , ( ) ( : ons distributi y probabilit Marginal ) , Pr( ) , ( : on distributi y probabilit Joint x x x x f x f x x f x f x X x X x x f

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CH Reilly UCF - IEMS - STA 3032 4 Example – joint distribution X 0 1 2 3 4 5 0 0 .05 .025 0 .025 0 .10 X 2 1 .20 .05 0 .30 0 0 .55 2 .10 0 0 0 .10 .15 .35 .30 .10 .025 .30 .125 .15
CH Reilly UCF - IEMS - STA 3032 5 Conditional distributions ) ( ) , ( ) | ( ) ( ) , ( ) | ( 1 1 2 1 1 2 2 2 2 2 1 2 1 1 x f x x f x x f x f x x f x x f = =

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CH Reilly UCF - IEMS - STA 3032 6 Expected value of a product of random variables ) ( E ) ( E ) , ( ) ( E 2 1 all all 2 1 2 1 2 1 1 2 X X x x f x x X X x x = ∑ ∑
CH Reilly UCF - IEMS - STA 3032 7 Independence of random variables ) ( E ) ( E ) ( E : ce independen Under , ), ( ) ( ) , ( : condition if) (only sufficient and (if) Necessary 2 1 2 1 2 1 2 2 1 1 2 1 X X X X x x x f x f x x f = 2200 =

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CH Reilly UCF - IEMS - STA 3032 8 Example (revisited) X 1 and X 2 are not independent. Pick any joint probability that is equal to 0. It is easy to show that this joint probability is not equal to the product of the corresponding column total of probabilities (marginal probability) and the corresponding row total of probabilities. We only need one counterexample to disprove independence.
CH Reilly UCF - IEMS - STA 3032 9 Covariance and correlation Covariance helps us measure the strength of the linear relationship between two random variables. Correlation is a unit-less measure of the strength of the linear relationship between two random variables.

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CH Reilly UCF - IEMS - STA 3032 10 Covariance and correlation (cont’d.) 1 1 ) ( E ) ( Var ) ( Var ) , ( Cov ) , ( Corr ) ( E ) ( E ) ( E ) , ( Cov 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 - - = = = - = ρ σ σ μ μ ρ X X X X X X X X X X X X X X X X X X
CH Reilly UCF - IEMS - STA 3032 11 Covariance and correlation (cont’d.) ed uncorrelat , 0 0 t independen , ed uncorrelat , 0 ) , ( Cov 0 ) , ( Cov t independen , 2 1 2 1 2 1 2 1 2 1 2 1 X X X X X X X X X X X X = = = = ρ ρ

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