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Unformatted text preview: Homework # 4 Solutions The George Washington University ApSc 116 1 / 11 Problem (Page 136, # 7) Suppose that the joint p.d.f. of X and Y is as follows: f ( x , y ) = 2 xe y for x 1 and < y < , otherwise. Are X and Y independent? Solution: Since f ( x , y ) = 0 outside a rectangle and f ( x , y ) can be factored as in Eq. (3.5.7), i.e., X and Y are independent iff f ( x , y ) = g 1 ( X ) g 2 ( Y ) , for some nonnegative functions g 1 and g 2 . Then inside the rectangle, using g 1 ( x ) = 2 x and g 2 ( y ) = exp ( y ), it follows that X and Y are independent. 2 / 11 Problem (Page 136, # 10) Suppose that a point ( X , Y ) is chosen at random from the circle S defined as follows S = { ( x , y ) : x 2 + y 2 1 } . (a) Determine the joint p.d.f. of X and Y , the marginal p.d.f. of X, and the marginal p.d.f. of Y . (b) Are X and Y independent? Solution: (a) f ( x , y ) is constant over the circle S . The area of S is units, and it follows that f ( x , y ) = 1 / inside S and f ( x , y ) = 0 outside S . Next, the possible values of x range from 1 to 1/ For any value x in this interval, f ( x , y ) > only for values of y between (1 x 2 ) 1 / 2 and (1 x 2 ) 2 ....
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 Fall '08
 TomMazzuchi

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