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Week5-Class4-Multivariate Distr.

# Week5-Class4-Multivariate Distr. - Multivariate...

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Multivariate Distributions Enrique Campos-N´ a˜nez The George Washington University ApSc 116

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Multivariate Data: Height, Weight, and Fat Motivating multivariate distributions

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Bivariate Distributions I The joint probability distribution of two r.v.’s is called a bivariate distribution . I The distribution is discrete if both r.v.’s are discrete. I Definition: The joint probability function (or joint p.f.) of X and Y is the function f such that ( x , y ) ∈ < 2 , f ( x , y ) = Pr ( X = x , Y = y ) . I If ( x , y ) is not one of the possible pairs of values of ( X , Y ) then f ( x , y ) = 0. Always, f ( x , y ) 0. I If the sequence { ( x i , y i ) : i = 1 , 2 , . . . } contains all possible values of ( X , Y ), then X i =1 f ( x i , y i ) = 1 . I Also, Pr ( X , Y ) A ⊂ < 2 = X ( x i , y i ) A f ( x i , y i ) .
Exercise Let X 1 , X 2 represent the number of points in two dice, and let X = min { X 1 , X 2 } , Y = max { X 1 , X 2 } . What is the joint probability function of X and Y ?

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Continuous Bivariate Distributions I We say that X and Y have a continuous joint distribution if there is a function f ( x , y ) 0 that is defined for all ( x , y ) ∈ < 2 and such that A ⊂ < 2 , Pr [( X , Y ) A ] = Z A Z f ( x , y ) dxdy . I The function f ( x , y ) is the joint probability density function or joint p.d.f., of X and Y . I In analogy to the univariate case, f ( x , y ) must satisfy f ( x , y ) 0 , -∞ < x , y < , and Z -∞ Z -∞ f ( x , y ) dxdy = 1 . I Note that if X and Y have a continuous joint distribution, then 1. Any point, or infinite sequence of points, in < 2 has a probability 0, and 2. Any one-dimensional curve in < 2 has probability 0.
Mixed Bivariate Distributions This covers all other cases. Example Select one person at random from members of the class and let X = 1 if the person is male, X = 2 if the person is female, and let Y be the weight in kilograms of the person.

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Bivariate Distribution Functions I Definition: The joint distribution function , or joint d.f., of r.v.’s X and Y is the function F such that ( x , y ) ∈ < 2 , F ( x , y ) Pr ( X x , Y y ) . Result. For all a b , and c d , we have Pr ( a < X b , c < Y d ) = F ( b , d ) - F ( a , d ) - F ( b , c ) + F ( a , c ) . a b c d
Bivariate Distribution Functions II Definition. The function F 1 ( x ) Pr ( X x ) = lim y →∞ Pr ( X x , Y y ) = lim y →∞ F ( x , y ) , and is called the marginal d.f. of X .

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