On the other hand if a b a b then a a so a v s

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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 Y|X (v|u 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.

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