09hw10 - EE 464 Mitra Homework 10 Due Monday 10am Spring...

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EE 464, Mitra Spring 2009 Homework 10 Due Monday, April 13, 2009, 10am This problem set investigates continues the examination of functions of random variables, as well as statistical relationships between multiple random variables and moments of multiple random variables. 1. Short Problems (a) If X 1 and X 2 are positive, identically distributed random variables with mean 1, show that E max { X 1 , X 2 } ≤ 2 . (b) Let Y = X + N , where X is a Bernoulli random variable with parameter p = 0 . 2 and N is Gaussian with zero mean and unit variance. Assume that X and N are independent. Determine the correlation coefficient ρ XY . Note this is akin to looking at the relationship between the transmitted signal and received signal in a binary communications system with coded data (hence p 6 = 0 . 5 ). 2. We have not done much on discrete random variables in lecture. Let X and Y and be two independent, geometric random variables: P [ X = k ] = p X (1 - p X ) k k = 0 , 1 , · · · P [ Y = k ] = p Y (1 - p Y ) k k = 0 , 1 , · · · (a) Determine the joint probability mass function of Z =
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