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Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.432 Stochastic Processes, Detection and Estimation Problem Set 1 Spring 2004 Issued: Tuesday, February 3, 2004 Due: Tuesday, February 10, 2004 Reading: For this problem set: Chapter 1 of course notes, through Section 1.6, Appendix 1.A Next: Chapter 2, through Section 2.5.1 Problem 1.1 A random variable x has probability distribution function P x ( x ) = [1 exp( 2 x )] u ( x ) where u ( · ) is the unit-step function. (a) Calculate the following probabilities: Pr [ x 1] , Pr [ x 2] , Pr [ x = 2] . (b) Find p x ( x ), the probability density function for x . (c) Let y be a random variable obtained from x as follows: 0 x < 2 y = . 1 x 2 Find p y ( y ), the probability density function for y . Problem 1.2 Let x and y be independent identically distributed random variables with common density function p ( ) = 1 0 0 1 otherwise . Let s = x + y . (a) Find and sketch p s ( s ). (b) Find and sketch p x | s ( x s ) vs. x with s viewed as a known parameter. | 1

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+ + + (c) The conditional mean of x given s = s is E [ x s = s ] = x p x | s ( x s ) dx. | −� | Find E [ x s = 0 . 5]. | (d) The conditional mean of x given s ( s viewed as a random
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