[RP]Lecture Note IV - 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) IV-1 C1002900 RP Lecture Handout IV: Two Random Variables Reading: Chapter 6 We will frequently deal with several random variables, and we will find it convenient to use vector notation to use vector notation in this case. Before we do that, however, let us recall some complete joint characterization of a pairs of random variables X and Y . 4.1. Joint distribution and Density The joint distribution function of X and Y is:    , ,, . XY F xy X xY y  The joint density is , , , ,. F fx y  2 (4.1) Let 2 be a region of the xy plane. Then, the probability that the point , is in is   , , P XY f xydxdy   (4.2) A special case of   . 42 is the inverse to   . 41 : . F f uvdudv     4.2. Marginal and Conditional Densities Marginal density:    , , XX Y f x y d y 
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Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) IV-2    , , YX Y fy f x y d x  The conditional density of X given that Yy is    , , . XY Y X Y fx y y x f x Conditional Expectation The conditional mean of X given is   XY y x f xydx   . This is a function of y . Therefore,    gY is a random variable whose expectation is given by        , , . YY Y xf x y dx X gyf ydy XY y f ydy xf x y dxdy xf x dx X         
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This note was uploaded on 06/10/2010 for the course ELECTRONIC C1002900 taught by Professor Hyungdongshin during the Spring '10 term at Kyung Hee.

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

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