h4soln

# h4soln - Homework 4 Solutions The George Washington...

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Homework # 4 Solutions The George Washington University ApSc 116 1 / 11

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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 0 x 1 and 0 < y < , 0 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 non-negative 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 π 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 ) > 0 only for values of y between - (1 - x 2 ) 1 / 2 and (1 - x 2 ) 2 . Hence for - 1 x 1, f 1 ( x ) = (1 - x 2 ) 1 / 2 - (1 - x 2 ) 1 / 2 1 π dy = 2 π (1 - x 2 ) 1 / 2 .

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• Fall '08
• TomMazzuchi
• Probability distribution, Probability theory, probability density function, George Washington University, marginal p.d.f.

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