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LectureNotes11 - Multivariate Probability Distributions:...

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Unformatted text preview: Multivariate Probability Distributions: Part II Cyr Emile MLAN, Ph.D. mlan@stat.uconn.edu Multivariate Probability Distributions: Part II p. 1/2 4 Introduction Text Reference : Introduction to Probability and Its Applications, Chapter 6. Reading Assignment : Sections 6.4-6.5, April 13-April 15 When we encounter problems that involve more than one random variable, we often combine the variables into a single function. We may be interested in the life lengths n electronic components within the same system. We now discuss expectation and covariances of a multivariate random variable. Multivariate Probability Distributions: Part II p. 2/2 4 Expected Values and Covariances Definition 6.11 : Let g ( X,Y ) be a function of 2 discrete random variables, X,Y , which have joint probability mass function, p ( x,y ) . Then the expected value of g ( X,Y ) is E bracketleftbig g ( X,Y ) bracketrightbig = summationdisplay x =- summationdisplay y =- g ( x,y ) p ( x,y ) (The sum is in fact over all values of ( x,y ) for which p ( x,y ) > ) If X,Y are continuous random variables with joint density function f ( x,y ) , then the expected value of g ( X,Y ) is E bracketleftbig g ( X,Y ) bracketrightbig = integraldisplay - integraldisplay - g ( x,y ) f ( x,y ) dxdy (The double integral is in fact over all values of ( x,y ) for which f ( x,y ) > Multivariate Probability Distributions: Part II p. 3/2 4 Expected Values and Covariances If g ( X, Y ) = XY , we have E bracketleftbig X Y bracketrightbig = summationdisplay x =- summationdisplay y =- xy p ( x,y ) , if discrete integraldisplay - integraldisplay - xy f ( x,y ) dxdy , if continuous If g ( X, Y ) = X k , we have E bracketleftbig X k bracketrightbig = summationdisplay x =- summationdisplay y =- x k p ( x,y ) , if discrete integraldisplay - integraldisplay - x k f ( x,y ) dxdy , if continuous If g ( X, Y ) = Y k , we have E bracketleftbig Y k bracketrightbig = summationdisplay x =- summationdisplay y =- y k p ( x,y ) , if discrete integraldisplay - integraldisplay - y k f ( x,y ) dxdy , if continuous Multivariate Probability Distributions: Part II p. 4/2 4 Expected Values and Covariances Example 6.9 : A process for producing an industrial chemical yields a product containing two types of impurities. For a specified sample from this process, let X denote the proportion of impurities in the sample and let Y denote the proportion of type 1 impurities among all impurities found. Suppose that the joint distribution of X and Y can be modeled by the following probability density function: f ( x,y ) = 2(1- x ) , if x 1 , y 1 , , otherwise ....
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LectureNotes11 - Multivariate Probability Distributions:...

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