Jacobian - GENERAL FUNCTIONS OF TWO RANDOM VARIABLES Random...

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1 GENERAL FUNCTIONS OF TWO RANDOM VARIABLES Random variables X and Y have their joint pdf , (,) XY f xy . Define another pair of random variables V and W as functions of X and Y as follows: 1 2 V g XY W g XY = = Assume h (.), the inverse function of g(.), exist so that 1 2 (, ) X hVW Y h VW = = Then , ,1 2 ( , (, ) ) | | VW f vw f h vw h vw J = where J is the Jacobian, 11 22 h vw J ∂∂ .
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2 Note. A similar formula holds for 3,4,5, …. random variables
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3 Raleigh Distribution X and Y are independent N (0,1). Define 22 1 tan V XY Y W X = = Then 2 2 () 0 1 0 2 2 v V W f v ve v Raleigh f w w Uniform π = = ≤< Homework: Derive the result. 2 , 1 Since and are independent (0,1), ( , ) 2 xy X Y N f e + = : cos sin Inverse Function X V W YV W = = cos cos (cos sin ) sin sin vw J v w wv ∂∂ = = += .
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4 ( ) ( ) 22 cos sin 2 ,, 1 ( , ) ( cos , sin ) 0 2 vw VW XY f v w f x v w y v w J e v for v π + = = = = 2 2 , 1 ( , ) 0, 0 2 2 v f v w ve for v w = ≤≤ 2 2 2 , 0 2 , 00 0 () (, ) 0 11 1 ( ) 0 2 2 2 v V vv W f v f v w dw ve for v v f w f v w dv ve dv e d for w ππ ∞∞ −− = =  = = = = ≤<   ∫∫
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This note was uploaded on 11/23/2010 for the course EE EE528 taught by Professor Majungsoo during the Spring '10 term at Korea Advanced Institute of Science and Technology.

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Jacobian - GENERAL FUNCTIONS OF TWO RANDOM VARIABLES Random...

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