Fundamentals of Probability and Statistics for Engineers

160 Fundamentals of Probability and Statistics for EngineersDetermine the jpdf of Y 1 and Y 2 , defined byY 1 ˆ X 1 ‡ X 2 ;Y 2 ˆ X1Y 1;and show that they are independent.5.29 The jpdf of X and Y is given byf XY …x; y† ˆ 1 2 2 exp …x 2 ‡ y 2 †2 2 ; … 1; 1† < …x; y† < …1; 1†:Determine the jpdf of R and and their respective marginal pdfs where2R ˆ ( X ‡ Y 2 ) 1/2 is the vector length and ˆ tan 1 (Y/X ) is the phase angle. AreR and independent?5.30 Show that an alternate formula for Equation (5.67) isf Y …y† ˆf X ‰g 1 …y†ŠjJ 0 j1 ;whereqg 1 …x† qg 1 …x† qg 1 …x†qx 1 qx 2 qx nJ 0 ˆ . . qg n …x† qg n …x† qg n …x† qx 1 qx 2 qx nis evaluated at x ˆ g 1 (y). Similar alternate forms hold for Equations (5.12), (5.24)and (5.68).TLFeBOOK