notes15 - Joint density of Order Statistics Suppose X1 X2...

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Joint density of Order Statistics Suppose X 1 ,X 2 ,...,X n are iid with pdf f ( x ). Let ( U 1 ,U 2 ,...,U n ) = ( X (1) ,X (2) ,...,X ( n ) ). Then f U 1 ,U 2 ,...,U n ( u 1 ,u 2 ,...,u n ) = n ! f ( u 1 ) f ( u 2 ) ··· f ( u n ) I ( u 1 < u 2 < ··· < u n ) . The order statistics are dependent ( not independent). The support of the joint density is the set { ( u 1 ,u 2 ,...,u n ) : u 1 < u 2 < · < u n } . This is not a Cartesian product set. We know that U i < U j for i < j . Thus, knowing U i tells us something about U j . So (intuitively) they must be depen- dent rv’s. 1
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Manipulating Joint Distributions Obtaining Marginal Density from Joint Density (continuous case) f W ( w ) = Z -∞ Z -∞ Z -∞ f W,X,Y,Z ( w,x,y,z ) dxdy dz f W,Y ( w,y ) = Z -∞ Z -∞ f W,X,Y,Z ( w,x,y,z ) dxdz f X,Y,Z ( x,y,z ) = Z -∞ f W,X,Y,Z ( w,x,y,z ) dw Conditional Densities f W | X,Y,Z ( w | x,y,z ) = f W,X,Y,Z ( w,x,y,z ) f X,Y,Z ( x,y,z ) f X,Z | W,Y ( x,z | w,y ) = f W,X,Y,Z
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notes15 - Joint density of Order Statistics Suppose X1 X2...

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