lect03 - Lecture Notes 3 Two Random Variables Joint,...

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Unformatted text preview: Lecture Notes 3 Two Random Variables Joint, Marginal, and Conditional PMFs Joint, Marginal, and Conditional CDFs, PDFs One Discrete and one Continuous Random Variables Signal Detection: MAP Rule Functions of Two Random Variables EE 278: Two Random Variables 3 1 Joint, Marginal, and Conditional PMFs Let X and Y be discrete random variables on the same probability space They are completely specified by their joint pmf : p X,Y ( x, y ) = P { X = x, Y = y } , x X , y Y By axioms of probability, X x X X y Y p X,Y ( x, y ) = 1 Example: Consider the pmf p ( x, y ) described by the following table x 1 2 . 5- 3 1 4 1 8 y- 1 1 8 1 4 2 1 8 1 8 EE 278: Two Random Variables 3 2 To find p X ( x ) , the marginal pmf of X , we use the law of total probability p X ( x ) = X y Y p ( x, y ) , x X The conditional pmf of X given Y = y is defined as p X | Y ( x | y ) = p X,Y ( x, y ) p Y ( y ) , p Y ( y ) 6 = 0 , x X Check that if p Y ( y ) 6 = 0 then p X | Y ( x | y ) is a pmf for X Chain rule : p X,Y ( x, y ) = p X ( x ) p Y | X ( y | x ) = p Y ( y ) p X | Y ( x | y ) X and Y are said to be independent if for every ( x, y ) X Y , p X,Y ( x, y ) = p X ( x ) p Y ( y ) , which is equivalent to p X | Y ( x | y ) = p X ( x ) , p Y ( y ) 6 = 0 , x X EE 278: Two Random Variables 3 3 Bayes Rule for PMFs Given p X ( x ) and p Y | X ( y | x ) for every ( x, y ) X Y , we can find p X | Y ( x | y ) : p X | Y ( x | y ) = p X,Y ( x, y ) p Y ( y ) = p X ( x ) p Y | X ( y | x ) p Y ( y ) = p Y | X ( y | x ) x X p X,Y ( x , y ) p X ( x ) = p Y | X ( y | x ) x X p Y | X ( y | x ) p X ( x ) p X ( x ) The final formula is entirely in terms of the known quantities p X ( x ) and p Y | X ( y | x ) EE 278: Two Random Variables 3 4 Example: Binary Symmetric Channel Consider the following binary communication channel X { , 1 } Y { , 1 } Z { , 1 } The bit sent is X Bern( p ) , p 1 , the noise is Z Bern( ) , . 5 , the bit received is Y = ( X + Z ) mod 2 = X Z , and X and Z are independent Find 1. p X | Y ( x | y ) 2. p Y ( y ) 3. P { X 6 = Y } , the probability of error EE 278: Two Random Variables 3 5 1. To find p X | Y ( x | y ) we use Bayes rule p X | Y ( x | y ) = p Y | X ( y | x ) x X p Y | X ( y | x ) p X ( x ) p X ( x ) We know p X ( x ) , but we need to find p Y | X ( y | x ) : p Y | X ( y | x ) = P { Y = y | X = x } = P { X Z = y | X = x } = P { x Z = y | X = x } = P { Z = y x | X = x } = P { Z = y x } since Z and X are independent = p Z ( y x ) Therefore p Y | X (0 | 0) = p Z (0 0) = p Z (0) = 1- p Y | X (0 | 1) = p Z (0 1) = p Z (1) = p Y | X (1 | 0) = p Z (1 0) = p Z (1) = p Y | X (1 | 1) = p Z (1 1) = p Z (0) = 1- EE 278: Two Random Variables 3 6 Plugging into the Bayes rule equation, we obtain p X | Y (0 | 0) = p Y | X (0 | 0) p Y | X (0 | 0) p X (0) + p Y | X (0 | 1) p X...
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lect03 - Lecture Notes 3 Two Random Variables Joint,...

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