hw1 soln

# hw1 soln - Dr Behnaam Aazhang ELEC 430 Department of...

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Dr. Behnaam Aazhang ELEC 430 Department of Electrical and Computer Engineering Rice University Due 17 Jan 2008 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 . 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 B C are also independent. Exercise 3. Probability Distributions Suppose X is a discrete random variable taking on values { 0 , 1 , 2 , . . . , n } with the following probability mass function: p X ( k ) = n ! k !( n - k )! θ k (1 - θ ) n - k k = { 0, 1, 2, . . . , n } 0 otherwise (3-1) with parameter θ [0 , 1] (a) Find the characteristic function of X (b) Find ¯ X and σ 2 X Hint: See problems 4.14 and 4.15 in Proakis and Salehi.

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• Spring '08
• Aazhang
• Volt, Probability theory, Probability mass function, Gaussian random variables, Proakis, Probability exercise

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