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HW2 - ENEE 620/FALL 2010 RANDOM PROCESSES IN COMMUNICATION...

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ENEE 620/FALL 2010 RANDOM PROCESSES IN COMMUNICATION AND CONTROL HOMEWORK # 2: Please work out the ten (10) problems stated below – GS refers to the text: Geoffrey R. Grimmett and David R. Stirzaker, Probability and Random Processes , Oxford University Press, Oxford (UK), 2001. With this in mind, Exercise 1.2-2 (GS) refers to Exercise 2 for Section 1.2 of GS while Problem 1-31 (GS) refers to Problem 31 for Chapter 1 in GS. Show work and explain reasoning. Three (3) problems, selected at random amongst these ten problems, will be marked. When not specified, an underlying probability triple (Ω , F , P ) is always assumed. 1. Recall that a collection of events A 1 , . . . , A n is said to be mutually independent if P [ i J A i ] = i J P [ A i ] for any subset J of { 1 , . . . , n } . Now consider a countably infinite collection { A n , n = 1 , 2 , . . . } of events. It is tempting to define the mutual independence of these events through the requirement P [ n J A n ] = n J P [ A n ] for any subset J of { 1 , 2 , . . . } . Explain why this is not a meaningful definition.
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