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lecture07 - ECE 5510: Random Processes Lecture Notes Fall...

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Unformatted text preview: ECE 5510: Random Processes Lecture Notes Fall 2008 Lecture 7 Today: (1) Method of Moments, cts r.v.s (Y&G 3.7), (2) Jaco- bian Method, cts. r.v.s Turn in application assignment 1, today (by midnight), on WebCT, or in the HW locker. HW 3 is due on Thursday at 5pm. Neal has office hours 1-3pm on Thu. 1 Method of Moments 1.1 Review Last time : For discrete r.v. X , P Y ( y ) = P [ { Y = y } ] = P [ { g ( X ) = y } ] = P bracketleftbig { X g- 1 ( y ) } bracketrightbig We did the Overtime example, which transforms the variable X (the number of rounds of overtime played) to T (the team that won), where we defined Team 1 as the team that went first, and Team 2 as the team that went second. See the mapping in Figure 1. When X was Geometric with parameter p , then T had the pmf, P T ( t ) = 1 2- p , t = 1 1- p 2- p , t = 2 , o.w. Defn: Many-to-One A function Y = g ( X ) is many-to-one if, for some value y , there is more than one value of x such that y = g ( x ), or equivalently, { g- 1 ( y ) } has more than one element. Defn: One-to-One A function Y = g ( X ) is one-to-one if, for every value y , there is exactly one value x such that y = g ( x ). Bottom line : know that the set { g- 1 ( y ) } can have either one, or many, elements. ECE 5510 Fall 2008 2 Figure 1: Overtime function T = g ( X ). Team 1 wins if overtime ends in an odd-numbered round, and Team 2 wins if overtime ends in an even-numbered round. Example: One-to-One Transform Let X be a discrete uniform r.v. on S X = { 1 , 2 , 3 } , and Y = g ( X ) = 2 X . What is P Y ( y )? Answer: We can see that S Y = { 2 , 4 , 6 } Then P Y ( y ) = P [ { Y = y } ] = P [ { 2 X = y } ] = P [ { X = y/ 2 } ] = P X ( y/ 2) = braceleftbigg 1 / 3 , y = 2 , 4 , 6 , o.w....
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lecture07 - ECE 5510: Random Processes Lecture Notes Fall...

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