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Unformatted text preview: CH Reilly UCF  IEMS  STA 3032 1 Bivariate, Sampling Distributions Chapters 5.10 & 6 Note Set 6 Spring 2007 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 CH Reilly UCF  IEMS  STA 3032 4 Example – joint distribution X 1 2 3 4 5 .05 .025 .025 .10 X 2 1 .20 .05 .30 .55 2 .10 .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 = = 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 = 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 unitless measure of the strength of the linear relationship between two random variables. 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...
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This note was uploaded on 08/22/2010 for the course STA 3032 taught by Professor Sapkota during the Spring '08 term at University of Central Florida.
 Spring '08
 Sapkota
 Statistics

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