501--10.23 - 4. If ( X, Y ) is cts, then F ( x, y ) =...

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Unformatted text preview: 4. If ( X, Y ) is cts, then F ( x, y ) = integraltext y- integraltext x- f ( u, v ) dudv , f ( x, y ) = 2 F ( x,y ) xy , provided that 2 F ( x,y ) xy exists. OW, f can be defined arbitrarily. 4.2. Conditional Distribution and Independence. Recall P ( A | B ) = P ( AB ) P ( B ) . If ( X, Y ) is discrete then denote f X | Y ( x | y ) = P (( X,Y )=( x,y )) P ( Y = y ) = f X,Y ( x,y ) f Y ( y ) or f ( x,y ) f Y ( y ) , but not f ( x,y ) f ( y ) . Definition. Let X and Y be random variables (vectors). The conditional density function of X given Y = y is f X | Y ( x | y ) = f X,Y ( x,y ) f Y ( y ) , where f X,Y is the joint density function of ( X, Y ). The conditional cdf of X given Y = y is F X | Y ( x | y ) = integraltext u x dF X | Y ( u | y ) def = braceleftbigg u x f X | Y ( u | y ) if X | Y = y is discrete integraltext u x f X | Y ( u | y ) du if X | Y = y is cts....
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