Lecture6 - BIVARIATE DISTRIBUTIONS Let x be a variable that...

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BIVARIATE DISTRIBUTIONS Let x be a variable that assumes the values { x 1 ,x 2 ,...,x n } . Then, a function that expresses the relative frequency of these values is called a univariate frequency function. It must be true that f ( x i ) 0 for all i and X i f ( x i )=1 . The following table provides a trivial example: x f ( x ) 1 0 . 25 1 0 . 75 Let x and y be variables that assume values in the the sets { x 1 2 n } and { y 1 ,y 2 ,...,y m } , respectively. Then the function f ( x i j ), which gives the relative frequencies of the occurrence of the pairs ( x i j ) is called a bivariate frequency function. It must be true that f ( x i j ) 0 for all i and X i X j f ( x i j . An example of a bivariate frequency table is as follows: y x 10 1 1 0 . 04 0 . 01 0 . 20 1 0 . 12 0 . 03 0 . 60 The values of f ( x i j ) are within the body of the table. The marginal frequency function of x gives the relative frequencies of the values of x i regardless of the values of y j with which they are associated; and it is deFned by f ( x i )= X j f ( x i j ); i =1 ,...,n. It follows that f ( x i ) 0 , and X i f ( x i X i X j f ( x i j , The marginal frequency function f ( y j ) is deFned analogously. The bivariate frequency table above provides examples of the two marginal frequency functions: f ( x = 1)=0 . 04 + 0 . 01 + 0 . 20=0 . 25 , f ( x =1)=0 . 12 + 0 . 03 + 0 . 60=0 . 75 . 1
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and f ( y = 1) = 0 . 04+0 . 12=0 . 16 , f ( y =0)=0 . 01+0 . 03=0 . 04 , f ( y =1)=0 . 20+0 . 60=0 . 80 . The conditional frequency function of x given y = y j gives the relative fre- quency of the values of x i in the subset of f ( x, y ) for which y = y j ; and it is given by f ( x i | y j )= f ( x i ,y j ) f ( y j ) . Observe that X i f ( x i | y j i f ( x i j ) f ( y j ) = f ( y j ) f ( y j ) =1 . An example based on the bivariate table is as follows: f ( x | y ) x f ( x | y = 1) f ( x | y =0) f ( x | y =1) 1 0 . 25 = (0 . 04 / 0 . 16) 0 . 25=(0 . 01 / 0 . 04) 0 . 25 = (0 . 20 / 0 . 80) 1 0 . 75 = (0 . 12 / 0 . 16) 0 . 75=(0 . 03 / 0 . 04) 0 . 75 = (0 . 60 / 0 . 80) We may say that x is independent of y if and only if the conditional distribution of x is the same for all values of y , as it is in this table. The conditional frequency functions of x are the same for all values of y if and only if they are all equal to the marginal frequency function of x . Proof. Suppose that f ( x i f ( x | y 1 ··· = f ( x | y m ). Then f ( x i X j f ( x i | y j ) f ( y j f ( x i ) X j f ( y j f ( x i ) , which is to say that f ( x i f ( x i ), Conversely, if the conditionals are all equal to the marginal, then they must be equal to each other. Also observe that, if f ( x i | y j f ( x i ) for all j and f ( y j | x i f ( y j ) for all i , then, equivalently, f ( x i j f ( x i | y j ) f ( y j f ( y j | x i ) f ( x i f ( x i ) f ( y j ) . The condition that f ( x i j f ( x i ) f ( y j ) constitutes an equivalent deFnition of the independence of x and y . We have been concerned, so far, with frequency functions. These provide the prototypes for bivariate probability mass functions and for bivariate probability density functions. The extension to probability mass functions is immediate. ±or the case of the density functions, we consider a two-dimensional space R 2 which is deFned as the set of all ordered pairs ( x, y ); −∞ <x ,y< , which correspond to the co-ordinates of the points in a plane of inFnite extent.
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Lecture6 - BIVARIATE DISTRIBUTIONS Let x be a variable that...

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