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

This preview shows pages 1–6. Sign up to view the full content.

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 ∂∂ .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2 Note. A similar formula holds for 3,4,5, …. random variables
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 ∂∂ = = += .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 ∞∞ −− = =  = = = = ≤<   ∫∫
5

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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.

Page1 / 8

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

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online