12 the total area of the product set a b is equal to a

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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.

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