Isye 2027

# The area of this region intersect s is thus the area

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Unformatted text preview: onal pmf of Y given X , pY |X (v |uo ), is deﬁned for all uo such that pX (uo ) > 0 by: pY |X (v |uo ) = P (Y = v |X = uo ) = pX,Y (uo , v ) pX (uo ). If pX (uo ) = 0 then pY |X (v |uo ) is undeﬁned. Example 4.2.1 Let (X, Y ) have the joint pmf given by Table 4.1. The graph of the pmf is shown in Figure 4.3. Find (a) the pmf of X , (b) the pmf of Y , (c) P {X = Y }, (d) P {X > Y }, (e) pY |X (v |2), which is a function of v . 122 CHAPTER 4. JOINTLY DISTRIBUTED RANDOM VARIABLES Table 4.1: A simple joint pmf. Y =3 Y =2 Y =1 0.1 0.1 0.2 0.3 X=2 X=1 0.2 0.1 X=3 v 3 2 p X,Y(u,v) 1 1 2 u 3 Figure 4.3: The graph of pX,Y . Solution: (a) The pmf of X is given by the column sums: pX (1) = 0.1, pX (2) = 0.3 + 0.2 + 0.1 = 0.6, pX (3) = 0.1 + 0.2 = 0.3. (b) The pmf of Y is given by the row sums: pY (1) = 0.3 + 0.1 = 0.4, pY (2) = 0.2 + 0.2 = 0.4, and pY (3) = 0.1 + 0.1 = 0.2. (c) P {X = Y } = pX,Y (1, 1) + pX,Y (2, 2) + pX,Y (3, 3) = 0 + 0.2 + 0 = 0.2. (d) P {X > Y } = pX,Y (2, 1) + pX,Y (3, 1) + pX,Y (3, 2) = 0.3...
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## This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Tech.

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