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12_DiscreteRVs-4_Joint_PMFs_packed

12_DiscreteRVs-4_Joint_PMFs_packed - 2 Discrete Random...

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2. Discrete Random Variables Part IV: Joint PMFs ECE 302 Spring 2012 Purdue University, School of ECE Prof. Ilya Pollak
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Joint PMF p X,Y of X and Y If X and Y are discrete random variables, their joint probability mass funcMon is defined as p X,Y (x,y) = P (X=x and Y=y) . The individual PMFs of X and Y are called the marginal PMFs and can be obtained from the joint PMF: p X ( x ) = p X , Y ( x , y ) y p Y ( y ) = p X , Y ( x , y ) x Ilya Pollak
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Example 2.9 (Fig 2.10) 0 1/20 1/20 1/20 1/20 2/20 3/20 1/20 1/20 2/20 3/20 1/20 1/20 1/20 1/20 0 y x Joint PMF p X,Y (x,y) Column sums: marginal PMF p X (x) E.g., p X (1) = P (X=1) = P (X=1, Y=1) + P (X=1, Y=2) + P (X=1, Y=3) + P (X=1, Y=4) = p X,Y (1,1) + p X,Y (1,2) + p X,Y (1,3) + p X,Y (1,4) = = 1/20 + 1/20 + 1/20 = 3/20 p X , Y (1, y ) y 1 1 2 2 3 3 4 4 Ilya Pollak
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Example 2.9 (Fig 2.10) 0 1/20 1/20 1/20 1/20 2/20 3/20 1/20 1/20 2/20 3/20 1/20 1/20 1/20 1/20 0 y x Joint PMF p X,Y (x,y) 3/20 7/20 7/20 3/20 Row sums: marginal PMF p Y (y) 3/20 6/20 8/20 3/20 Column sums: marginal PMF p X (x) p X (1) = 3/20; p X (2) = 6/20; p X (3) = 8/20; p X (4) = 3/20; p Y (1) = 3/20; p Y (2) = 7/20; p Y (3) = 7/20; p Y (4) = 3/20; 1 1 2 2 3 3 4 4 Ilya Pollak
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Joint PMF for more than two discrete random variables p X 1 , X 2 , , X n ( x 1 , x 2 , , x n ) = P ( X 1 = x 1 , X 2 = x 2 , , X n = x n ) The marginal PMF for each X i can be obtained from the joint PMF by summing over all the x j 's other than x i , for example, p X 1 ( x 1 ) = p X 1 , X 2 , , X n ( x 1 , x 2 , , x n ) x n x 3 x 2 Ilya Pollak
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FuncMons of many discrete random variables Let Y = g ( X 1 , X 2 , , X n ). Then p Y ( y ) = p X 1 , X 2 , , X n ( x 1 , x 2 , , x n ) {( x 1 , , x n )| g ( x 1 , , x n ) = y } E [ g ( X 1 , X 2 , , X n )] = g ( x 1 , , x n ) p X 1 , , X n ( x 1 , , x n ) x 1 , , x n If g ( X 1 , X 2 , , X n ) = a 0 + a 1 X 1 + a 2 X 2 + + a n X n , then E [ g ( X 1 , X 2 , , X n )] = a 0 + a 1 E [ X 1 ] + a 2 E [ X 2 ] + + a n E [ X n ] Ilya Pollak
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Example 2.9 (conMnued) 0 1/20 1/20 1/20 1/20 2/20 3/20 1/20 1/20 2/20 3/20 1/20 1/20 1/20 1/20 0 y x Joint PMF p X,Y (x,y) 3/20 7/20 7/20 3/20 3/20 6/20 8/20 3/20 1 1 2 2 3 3 4 4 Z = X +2 Y (a) Find p Z (k) .
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