hw1sol - Dr Ashu Sabharwal ELEC 430 Department of...

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Unformatted text preview: Dr. Ashu Sabharwal ELEC 430 Department of Electrical and Computer Engineering Rice University Due 18 Jan 2007 HOMEWORK 1 — Probability Exercise 1. Change of Variables The current I in a semiconductor diode is related to the voltage V by the relation I = e V- 1. If V is a random variable with density function f V ( x ) = 1 2 e-| x | for-∞ < x < ∞ , find f I ( y ); the density function of I . Here we use the change of variables equation: f I ( y ) = f v ( g- 1 ( y )) | d dy g- 1 ( y ) | with y = g ( x ) = e x- 1 giving us g- 1 ( y ) = ln ( y- 1) and | d dy g- 1 ( y ) | = | 1 y + 1 | so f I ( y ) = 1 2 e-| ln ( y +1) | | 1 y + 1 | Since y = e x- 1 , y ≥ - 1 ∀ x and if we note the sign change in ln ( y + 1) when y = 0 we have the following two cases: for y ≥ f I ( y ) = 1 2( y + 1) e- ln ( y +1) and for- 1 ≤ y < f I ( y ) = 1 2( y + 1) e ln ( y +1) f I ( y ) = 1 2 Thus our density for I is: f I ( y ) = 1 2( y +1) e- ln ( y +1) y ≥ 1 2- 1 ≤ y < y <- 1 ELEC 430 Homework 1 Spring 2007 1-→ A nswer Exercise 2. Axioms of Probability (a) Show that if A ∩ B = { } then P [ A ] ≤ P B C (b) Show that for any A,B, C we have P [ A ∪ B ∪ C ] = P [ A ] + P [ B ] + P [ C ]- P [ A ∩ B ]- P [ A ∩ C ]- P [ B ∩ C ] + P [ A ∩ B ∩ C ] (c) Show that if A and B are independent then P A ∩ B C = P [ A ] P B C which means that A and...
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hw1sol - Dr Ashu Sabharwal ELEC 430 Department of...

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