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Unformatted text preview: Outline Models for Joint Distributions (Continued) Lecture 16 Chapter 4: Multivariate Variables and Their Distribution M. George Akritas M. George Akritas Lecture 16 Chapter 4: Multivariate Variables and Their Distribu Outline Models for Joint Distributions (Continued) Models for Joint Distributions (Continued) The Bivariate Normal Distribution Multinomial Distribution M. George Akritas Lecture 16 Chapter 4: Multivariate Variables and Their Distribu Outline Models for Joint Distributions (Continued) The Bivariate Normal Distribution Multinomial Distribution If we take the normal simple linear regression model and specify, in addition, that the explanatory variable X is normally distributed, then X and Y are said to have the bivariate normal joint distribution. M. George Akritas Lecture 16 Chapter 4: Multivariate Variables and Their Distribu Outline Models for Joint Distributions (Continued) The Bivariate Normal Distribution Multinomial Distribution If we take the normal simple linear regression model and specify, in addition, that the explanatory variable X is normally distributed, then X and Y are said to have the bivariate normal joint distribution. Definition If the joint distribution of ( X , Y ) is specified by the assumptions that the conditional distribution of Y given X = x follows the normal linear regression model, and X has a normal distribution, i.e. if Y  X = x N ( + 1 ( x X ) , 2 ) , and X N ( X , 2 X ) , then ( X , Y ) is said to have a bivariate normal distribution . M. George Akritas Lecture 16 Chapter 4: Multivariate Variables and Their Distribu Outline Models for Joint Distributions (Continued) The Bivariate Normal Distribution Multinomial Distribution It follows that the joint pdf of ( X , Y ) is f X , Y ( x , y ) = 1 p 2 2 exp ( y  1 ( x X )) 2 2 2 1 q 2 2 X exp ( x X ) 2 2 2 X M. George Akritas Lecture 16 Chapter 4: Multivariate Variables and Their Distribu Outline Models for Joint Distributions (Continued) The Bivariate Normal Distribution Multinomial Distribution It follows that the joint pdf of ( X , Y ) is f X , Y ( x , y ) = 1 p 2 2 exp ( y  1 ( x X )) 2 2 2 1 q 2 2 X exp ( x X ) 2 2 2 X It can be shown that the following is true Proposition If ( X , Y ) have a bivariate normal distribution, then the marginal distribution of Y is also normal with mean Y and variance 2 Y = 2 + 2 1 2 X . M. George Akritas Lecture 16 Chapter 4: Multivariate Variables and Their Distribu Outline Models for Joint Distributions (Continued) The Bivariate Normal Distribution Multinomial Distribution I A bivariate normal distribution is completely specified by X , Y , 2 X , 2 Y and XY ....
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 Fall '00
 Akritas

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