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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 λz e μz = 1 − e ( λ + μ ) z Hence, Z has an exponential distribution with parameter λ + μ 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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