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501--10.23

# 501--10.23 - y 2 x 2 F(x,y F(x,y 4 If(X Y is cts then F(x y...

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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 ) ∂x∂y , provided that 2 F ( x,y ) ∂x∂y 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. The conditional expectation of X | Y = y is E ( X | Y = y ) = integraltext xdF X | Y ( x | y ). The conditional variance of X | Y = y is V ( X | Y = y ) = integraltext ( u E ( X | Y = y )) 2 dF X | Y ( u | y ). X and Y are independent ( X Y ) if F X,Y ( x,y ) = F X ( x ) F Y ( y ) ( x,y ). OW, we say X and Y are dependent ( X negationslash⊥ Y ).
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