Fortunately in those instances independence is also

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Unformatted text preview: X = uo , where 0 ≤ uo < 1. bounded by the pdf along u ≡ uo is a two-dimensional triangular region. Uniform joint pdfs A simple class of joint distributions are the distributions that are uniform over some set. Let S be a subset of the plane with ﬁnite area. Then the uniform distribution over S is the one with the following pdf: 1 area of S if (u, v ) ∈ S fX,Y (u, v ) = 0 else. If A is a subset of the plane, then P {(X, Y ) ∈ A} = area of A ∩ S . area of S (4.9) The support of the uniform pdf over S is simply S itself. The next two examples concern uniform distributions over two diﬀerent subsets of R2 . Example 4.3.2 Suppose (X, Y ) is uniformly distributed over the unit circle. That is, take S = {(u, v ) : u2 + v 2 = 1}. Since the area of S is π, the joint pdf is given by fX,Y (u, v ) = 1 π 0 if u2 + v 2 ≤ 1 else. The pdf and its support are pictured in Figure 4.6. The three dimensional region under the pdf 1 (and above the u − v plane) is a cylinder centered at t...
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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.

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