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Unformatted text preview: itional density fX Y (v uo ) is undeﬁned if uo  ≥ 1, so suppose uo  < 1. Then
fX Y (v uo ) = 1 2 1−u2
o
π = √1 if − 0 √π else. 2 1−u2
o 1 − u2 ≤ v ≤
o 1 − u2
o That is, if uo  < 1, then given X = uo , Y is uniformly distributed over the interval − 1 − u2 , 1 − u2 .
o
o
This makes sense geometrically–a slice through the cylindrically shaped region under the joint pdf
is a rectangle. Example 4.3.3 Here’s a second example of a uniform distribution over a set in the plane. Suppose
(X, Y ) is uniformly distributed over the set S = {(u, v ) : 0 ≤ u ≤ 1, 0 ≤ v ≤ 1, max{u, v } ≥ 0.5}.
The pdf and its support set S are shown in Figure 4.7. Since the area of S is 3/4, the pdf is:
fX,Y (u, v ) =
Find the marginal and conditional pdfs. 4
3 if (u, v ) ∈ S
0 else. 128 CHAPTER 4. JOINTLY DISTRIBUTED RANDOM VARIABLES v
f 1 v (u,v) X,Y 0.5 0.5 4/3 support
S
u u
0.5 1 0.5 1 Figure 4.7: The pdf fX,Y and its support, S , for (X, Y ) uniformly distribute...
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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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