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tranformation_example

tranformation_example - x 2< ∞ → y 1 y 2 = 0 y 1< y...

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Example Let X 1 and X 2 be independent random variables, each with p.d.f. Consider Y 1 = X 1 X 2 and Y 2 = X 1 + X 2 find the joint p.d.f. of Y 1 and Y 2 . , ) ( x e x f - = < < x 0
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Example Their joint p.d.f. is Hence, The Jacobian is equal to 2 1 ) ( ) ( 2 1 x x e x f x f - - = < < < < 2 1 0 , 0 x x 2 2 1 1 y y x + = 2 1 2 2 y y x - = 2 1 2 1 2 1 2 1 2 1 = = - J
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Example Note that x 1 = 0, 0 <
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Unformatted text preview: x 2 < ∞; → y 1 + y 2 = 0, y 1 < y 2 ; x 2 = 0, 0 ≤ x 1 < ∞; → y 2 = y 1 , y 2 > - y 1 ; The joint p.d.f. of Y 1 and Y 2 is , 2 1 ) , ( 2 2 1 y e y y g-= ∞ < < < <-2 2 1 2 , y y y y...
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