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.
 Spring '08
 Zahrn
 The Land

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