12_Sim_Hw2_Sol

12_Sim_Hw2_Sol - IEOR 4404 Assignment #2 Solutions...

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Unformatted text preview: IEOR 4404 Assignment #2 Solutions Simulation February 1, 2012 Prof. Mariana Olvera-Cravioto Page 1 of 2 Assignment #2 Solutions 1. The CDF of Z is: F Z ( z ) = P ( Z z ) = 1 P (min { X, Y } z ) = 1 P ( X z, Y z ) = 1 P ( X z ) P ( Y z ) by independence = 1 e z e z = 1 e ( + ) z Hence, Z has an exponential distribution with parameter + 2. (a) Let denote the triangle formed by the points (0 , 0), (1 , 0) and (0 , 2). Since X and Y are uniformly distributed over , it follows that the joint PDF of X and Y is given by f X,Y ( x, y ) = braceleftBigg 1 Area ( ) if ( x, y ) otherwise. But Area( ) = 1 and can be described as = { ( x, y ) : x , y , 2 x + y 2 } . Therefore f X,Y ( x, y ) = braceleftBigg 1 if x , y , 2 x + y 2 otherwise. (b) The marginal PDF of X is given by f X ( x ) = integraldisplay f X,Y ( x, y ) dy = braceleftBigg integraltext 2 x +2 1 dy if 0...
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12_Sim_Hw2_Sol - IEOR 4404 Assignment #2 Solutions...

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