Lecture9 - Ex: 2 ( x 2 + y 2 + x 2 y 2 + 1) Are x and y...

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•1 Lecture 9 Ex: Are x and y independent? First factor out as 22 2 2 2 1 (, ) (1 ) xy fx y xyx y π = ++ + xy f xy 2 2 1 [( 1 )( 1 ) ] xy y xyy = +++ ) 1 ( 1 ) 1 ( 1 2 2 x y + + = 11 () ) ) x f xd y yx ππ −∞ =⋅ ++ ) 1 ( 1 )) 2 ( 2 ( 1 ) 1 ( 1 | tan 1 ) 1 ( 1 2 2 1 2 x x y x + = + = + = Can similarly obtain => independent ) 1 ( 1 ) ( 2 y y f y + = () () xy x y f f xf y = 2 ) , ( R y x Independence Tests ± Graphical test of independence From previous discussion, x and y are independent then If we have equality for each y, then: Corollary: x and y are independent RV’s if Corollary: x and y are independent RV’s if (|) xx Fxy Fx = R y x , 12 (| ) Fxy = R y y x 2 1 , , f xy = R y y x 2 1 , ,
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•2 Question: if we transform two independent RV’s by g(.) and h(.) , what can we say about g(x) and h(y)? ± Independence of functions of RV’s Theorem : Let x and y be two independent RV’s , let V=g(x) and W=h(y), then V and W are independent RV’s. ± Transformation of a Random vector: We distinguish 3 cases of interest: 1. One function of two RV’s : () Zx , y g = 2. Two functions of two RV’s: 3. More than two functions of two RV’s REMARK: It is important to note( in fact convince yourself) that uncorrelation does not imply independence. 11 zx , y g = 22 , y g = , y ii g = N i " , 1 = Example: Let(x,y) be a random two-tuple vector. The PDF of the vector is: X Y 1 4 3 2 (, ) xy f xy = 2 2 2 r y x C < + 2 2 2 0 r y x > + i.e. If x=0, y can take any value between –r and r with equal prob. If x=r, y can only be equal to zero. This clearly establishes the dependence between x and y . 2 1 r C π =
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•3 If we now compute Hence establishing non-correlation of (x,y) while still dependent. ± Transformation of vectors: Ex: suppose we have , what is the PDF/distribution of Z when given () ( , ) xy x,y E xyf x y dxdy = ∫∫ 0 1 2 = = ∫∫ dxdy xy r C π zx , y g = (, ) XY f xy ( ) ( ( ) ) Z , y Gz P z Pg z = ≤= , y g = By computation : ( ( ) ) , y P zP g z (( ) ) x,y P D =∈ xy D f x y dxdy = X Z Y D Ex2: Z=XY Question: what is the distribution of such a transformation?
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This note was uploaded on 09/24/2009 for the course ECE 514 taught by Professor Krim during the Fall '08 term at N.C. State.

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Lecture9 - Ex: 2 ( x 2 + y 2 + x 2 y 2 + 1) Are x and y...

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