This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 π units, and it follows that f ( x , y ) = 1 /π 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 ....
View
Full
Document
This note was uploaded on 01/02/2010 for the course APSC 116 taught by Professor Tommazzuchi during the Fall '08 term at GWU.
 Fall '08
 TomMazzuchi

Click to edit the document details