hw1 - EEcfw_564, Homework 1 Due Monday, September 10, 2001...

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EE{564, Fall 2009 Homework 1 Due Monday, September 10, 2001 These problems have two main objectives: to recall key skills and tools from probability theory and random processes and to develop a feel for comparing experimental/computer results with calculated quantities. 1. This problem addresses a method for converting two independent uniform random variables into two independent Gaussian random variables. Consider the independent uniformly dis- tributed random variables X 1 and X 2 : p X 1 ( x ) = p X 2 ( x ) = 8 > < > : 1 x 2 (0 ; 1) 0 else The purpose of this problem is to demonstrate that the following are independent Gaussian random variables: Y 1 = q 2 ln ( X 1 ) cos(2 2 ) Y 2 = q 2 ln ( X 1 ) sin(2 2 ) (a) Determine the following, p X 1 ;X 2 ( x 1 ;x 2 ) , E f Y 1 g , E f Y 2 g , and E f Y 1 Y 2 g . (b) Consider the random variable, R = q 2 ln ( X 1 ) . Determine the pdf p R ( r ) of this random variable and its mean. (c)
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hw1 - EEcfw_564, Homework 1 Due Monday, September 10, 2001...

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