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Unformatted text preview: Solutions to PS 7 Probability for Engineering Applications (PEA), ECSE 4500, Spring 2010 3.28 X and Y are given to be jointly Gaussian random variables. Consider two other random variables given by the relation Z = g ( X, Y ) = X 2 + Y 2 , and W = h ( X ) = X. The first step to find the joint pdf of Z and W is to find the roots of the equations z = g ( x, y ) and w = h ( x ). The real roots of these wo equations are give by R 1 = ( x 1 , y 1 ) and R 2 = ( x 2 , y 2 ). R 1 : x 1 = w, y 1 = z w 2 . R 2 : x 2 = w, y 2 = z w 2 . Note that  w  < z , and z > 0. The Jacobian is given by J = vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle g x g y h x h y vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle = vextendsingle vextendsingle vextendsingle vextendsingle 2 x 2 y 1 vextendsingle vextendsingle vextendsingle vextendsingle = 2 y. The magnitude of the Jacobian at the two roots are  J 1  =  2 y 1  = 2 z w 2 and  J 2  =  2 y 2  = 2 z w 2 . Now the joint pdf of Z and W is given by f ZW ( z, w ) = 2 summationdisplay i =1 f XY ( x i , y i ) J i = 1 2 z w 2 bracketleftBig f XY ( w, z w 2 ) + f XY...
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 Spring '08
 WOODS

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