[RP]Lecture Note III

# [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    : xgx c is a Borel subset of . Let   Yg X  . Then, Y maps to and Y is a random variable, denoted by YgX . X g X gX Y   Example 3.1: Let , X    2 23 and YX 2 . Find   Y fy . Since   PY  01 ,   Y Fy 0 for y 0. For y 0, we have     . Y Fy PX y Py Xy yy X P GG            2 22 2 33 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 Gx e 2 2 1 2 , we get exp exp , . YY d fy Fy dy yy y y             22 1 0 24 6 6 Example 3.2 (Hard Limiter): Let if x gx x  10 and YgX . Then , Y takes the values 1 with       . X X PY PX F F       0 1 0 o x   X Fx x   g x 1 1 X Y o y   Y 1 1 1 y   Y f y 1 1   X F 0 X F 0   X F d dy
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  , 01. Le t 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   gU . An ap- propriate function g is given by the inverse function of F . Although F may not be strictly increasing, a suitable version of F 1

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

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