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Unformatted text preview: X (u)fY (v ). So if fX,Y (a, b) > 0 and fX,Y (c, d) > 0, it must
be that fX (a), fY (b), fX (c), and fY (d) are all strictly positive, so fX,Y (a, d) > 0 and fX,Y (b, d) > 0.
That is, the support of fX,Y satisﬁes the condition of Proposition 4.4.3, and therefore it is a product
set. Corollary 4.4.5 Suppose (X, Y ) is uniformly distributed over a set S in the plane. Then X and
Y are independent if and only if S is a product set.
Proof.(if) If X and Y are independent, the set S must be a product set by Proposition 4.4.4.
(only if ) Conversely, if S is a product set, then S = A × B for some sets A and B , and S  = AB .
Thus, the joint density of X and Y is given by
fX,Y (u, v ) = 1
AB  0 u ∈ A, v ∈ B
else. The standard integral formulas for marginal pdfs, (4.4) and (4.5), imply that X is uniformly
distributed over A, Y is uniformly distributed over B. So fX,Y (u.v ) = fX (u)fY (v ) for all u, v.
Therefore, X is independent of Y. Example 4.4.6 Decide whether X and Y are independent for each of the following three pdfs:
C u2 v 2 u, v ≥ 0, u + v ≤ 1
for an appropriate choice of C.
(a) fX,Y (u, v ) =
0...
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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 Institute of Technology.
 Spring '08
 Zahrn
 The Land

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