Chapter 5: JOINT PROBABILITY DISTRIBUTIONS Part 3: The ...

Chapter 5: JOINT PROBABILITYDISTRIBUTIONSPart 3:The Bivariate NormalSection Functions of Random VariablesSection 5-41 The bivariate normal is kind of nifty The marginal DISTRIBUTIONS ofXandYareboth univariate normal DISTRIBUTIONS . The conditional distribution ofYgivenXisa normal distribution . The conditional distribution ofXgivenYisa normal distribution . Linear combinations ofXandY(such asZ= 2X+ 4Y) follow a normal distribution . It s normal almost any way you slice Bivariate Normal ProbabilityDensity FunctionThe parameters: X, Y, X, Y, fXY(x,y) =12 X Y (1 2) exp{ 12(1 2)[(x X)2 2X 2 (x X)(y Y) X Y+(y Y)2 2Y]}for < x < and < y < , withparameters X>0 , Y>0 , < X< , < Y< ,and 1< < is the correlation other parameters are the needed parame-ters for the marginal DISTRIBUTIONS Bivariate NormalWhenXandYareindependent, the con-tour plot of the JOINT distribution looks like con-centric circles (or)

Example: From book problem 5-54. Assume X and Y have a bivariate normal distribution with.. X= 120;˙X= 5 Y = 100;˙Y = 2 ˆ= 0:6 Determine: (i) Marginal probability distribution of X. (ii) Conditional probability distribution of Y given that X= 125. 10

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  Chapter, Distribution, Chapter 5

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