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Unformatted text preview: Outline Parameters of a Multivariate Distribution Lecture 15 Chapter 4: Multivariate Variables and Their Distribution Michael Akritas Michael Akritas Lecture 15 Chapter 4: Multivariate Variables and Their Distribu Outline Parameters of a Multivariate Distribution Parameters of a Multivariate Distribution The Regression Function Covariance Pearsons (or Linear) Correlation Coefficient Spearmans (or Monotone) Correlation Michael Akritas Lecture 15 Chapter 4: Multivariate Variables and Their Distribu Outline Parameters of a Multivariate Distribution The Regression Function Covariance Pearsons (or Linear) Correlation Coefficient Spearmans (or Monotone) Correlation Michael Akritas Lecture 15 Chapter 4: Multivariate Variables and Their Distribu Outline Parameters of a Multivariate Distribution The Regression Function Covariance Pearsons (or Linear) Correlation Coefficient Spearmans (or Monotone) Correlation I For a bivariate r.v. ( X , Y ), the regression function of Y on X shows how the conditional mean of Y changes with the observed value of X . More precisely, Definition The conditional expected value of Y given that X = x , i.e. Y  X ( x ) = E ( Y  X = x ) , when considered as a function of x , is called the regression function of Y on X . Michael Akritas Lecture 15 Chapter 4: Multivariate Variables and Their Distribu Outline Parameters of a Multivariate Distribution The Regression Function Covariance Pearsons (or Linear) Correlation Coefficient Spearmans (or Monotone) Correlation I For a bivariate r.v. ( X , Y ), the regression function of Y on X shows how the conditional mean of Y changes with the observed value of X . More precisely, Definition The conditional expected value of Y given that X = x , i.e. Y  X ( x ) = E ( Y  X = x ) , when considered as a function of x , is called the regression function of Y on X . I The regression function of Y on X is the fundamental ingredient for predicting Y from and observed value of X . Michael Akritas Lecture 15 Chapter 4: Multivariate Variables and Their Distribu Outline Parameters of a Multivariate Distribution The Regression Function Covariance Pearsons (or Linear) Correlation Coefficient Spearmans (or Monotone) Correlation Example In Lecture 14 we discussed the example where X = amount of drug administered, Y = number of tumors developed. In this example, y 1 2 p Y  X =0 ( y ) .97 .0225 .0075 p Y  X =1 ( y ) .97 .02 .01 p Y  X =2 ( y ) .9 .08 .02 Find the regression function of Y on X . Michael Akritas Lecture 15 Chapter 4: Multivariate Variables and Their Distribu Outline Parameters of a Multivariate Distribution The Regression Function Covariance Pearsons (or Linear) Correlation Coefficient Spearmans (or Monotone) Correlation Example In Lecture 14 we discussed the example where X = amount of drug administered, Y = number of tumors developed. In this example, y 1 2 p Y  X =0 ( y ) .97 .0225 .0075 p Y  X =1 ( y ) .97 .02 .01 p Y  X =2 ( y ) .9 .08 .02....
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This note was uploaded on 03/19/2009 for the course STAT 401 taught by Professor Akritas during the Spring '00 term at Pennsylvania State University, University Park.
 Spring '00
 Akritas

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