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Unformatted text preview: d over S.
Solution:
∞ fX (uo ) = fX,Y (uo , v )dv =
−∞ 14
0.5 3 dv = 2
3 0 ≤ uo < 0.5 14
0 3 dv = 4
3 0.5 ≤ uo ≤ 1 0 else. The graph of fX is shown in Figure 4.8. By symmetry, fY is equal to fX . f f (u) 0< uo <0.5 X 4/3 YX (vu o )
0.5 < uo < 1 2 2/3
u
0 0.5 1
v 1
0 0.5 1 v
0 0.5 1 Figure 4.8: The marginal and conditional pdfs for (X, Y ) uniformly distributed over S.
The conditional density fY X (v uo ) is undeﬁned if uo < 0 or uo > 1. It is well deﬁned for
0 ≤ uo ≤ 1. Since fX (uo ) has diﬀerent values for uo < 0.5 and uo ≥ 0.5, we consider these two cases
separately. The result is
fY X (v uo ) =
fY X (v uo ) = 2 0.5 ≤ v ≤ 1
0 else
1 0≤v≤1
0 else for 0 ≤ uo < 0.5,
for 0.5 ≤ uo ≤ 1. That is, if 0 ≤ uo < 0.5, the conditional distribution of Y given X = uo is the uniform distribution
over the interval [0.5, 1], And if 0.5 ≤ uo ≤ 1, the conditional distribution of Y given X = uo is the
uniform distribution over the interval [0, 1]. The conditional dist...
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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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