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[RP]Lecture Note III - Kyung Hee University Department of...

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Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) III-1 C1002900 RP Lecture Handout III: Functions of a Random Variable Reading: Chapter 5 3.1. Distribution of g X Let X be a random variable and g be a function mapping to . Specifically, sup- pose for any constant c that   : x g x c is a Borel subset of . Let Y g X   . Then, Y maps to and Y is a random variable, denoted by Y g X . X g X g X Y Example 3.1: Let , X   2 2 3 and Y X 2 . Find   Y f y . Since P Y 0 1 ,   Y F y 0 for y 0 . For y 0 , we have   . Y F y P X y P y X y y y X P y y G G 2 2 2 2 3 3 3 2 2 3 3
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Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) III-2 Since   ' x G x e 2 2 1 2 , we get     exp exp , . Y Y d f y F y dy y y y y 2 2 2 2 1 0 24 6 6 Example 3.2 (Hard Limiter): Let   , if , if x g x x 1 0 1 0 and Y g X . Then , Y takes the values 1 with     . X X P Y P X F P Y P X F     1 0 0 1 0 1 0 o x   X F x x   g x 1 1 X Y o y   Y F y 1 1 1 y   Y f y 1 1   X F 0   X F 0   X F 1 0 d dy
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Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) III-3 Inverse Problem : An important step in many computer simulations of random systems is to generate a random variable with a specified CDF, by applying a function to a random variable that is uniformly distributed on the interval , 0 1 . Let F be a function satisfying the three properties required of a CDF, and let U be uniformly distributed over the inter- val , 0 1 . The problem is to find a function g such that F is the CDF of g U . An ap- propriate function g
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