42 x and y are independent if and only if the

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Unformatted text preview: he origin with height π . Note that the volume of the region is 1, as required. (a) Find P {(X, Y ) ∈ A} for A = {(u, v ) : u ≥ 0, v ≥ 0}. (b) Find P {X 2 + Y 2 ≤ r2 } for r ≥ 0. (c) Find the pdf of X . (d) Find the conditional pdf of Y given X. 4.3. JOINT PROBABILITY DENSITY FUNCTIONS 127 v f X,Y 1 (u,v) support of f X,Y 1 (u,v) !1 1 u u 1 v !1 Figure 4.6: The pdf for X, Y uniformly distributed over the unit circle in R2 Solution: (a) The region A ∩ S is a quarter of the support, S , so by symmetry, P {(X, Y ) ∈ A} = 1/4. (b) If 0 ≤ r ≤ 1, the region {(u, v ) : u2 + v 2 ≤ r2 } is a disk of radius r contained in S. The area of this region intersect S is thus the area of this region, which is πr2 . Dividing this by the area of S yields: P {X 2 + Y 2 ≤ r} = r2 , for 0 ≤ r ≤ 1. If r > 1, the region {(u, v ) : u2 + v 2 ≤ r2 } contains S , so P {X 2 + Y 2 ≤ r} = 1 for r > 1. (c) The marginal, fX , is given by √ √2 1−u2 1 ∞ √ o 2 π dv = 2 1−uo if |uo | ≤ 1 π − 1−uo fX,Y (uo , v )dv = fX (uo ) = 0 −∞ else. (d) The cond...
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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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