Assignment02Sol

Assignment02Sol - Columbia University IEOR 4404: Simulation...

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Columbia University IEOR 4404: Simulation Fall 2009 Solution to Assignment 2 1. Use simulation to approximate the following integrals. Compare your estimate with the exact answer if known. θ 1 = Z 0 ( - 2 x - x 2 )(1 + x + x 2 ) - 2 dx (1) θ 2 = Z 0 Z x 0 e - x y dy dx (2) [Hint: Let I y ( x ) = 1 if y < x 0 if y x and use this function to equate the integral to one in which both terms go from 0 to .] Solution: By making the substitution y = 1 x +1 , dx = - 1 y 2 dy in Equation (1), we obtain: θ 1 = Z 1 0 y 2 - 1 ( y - 1 - y 2 ) 2 dy (3) Therefore, if U 1 ,...,U k are independent uniform (0 , 1) random variables, 1 k k X i =1 U 2 i - 1 ( U i - 1 - U 2 i ) 2 is the Monte Carlo approximation of θ 1 . To approximate the second integral, we use the hint: θ 2 = Z 0 Z 0 e - x yI y ( x ) dy dx Then, we apply the following substitution (remember that the substitution function has to be injective!) ( u,v ) = φ ( x,y ) = ( 1 x +1 , 1 y +1 ). It is trivial to see that φ is injective and the determinant of its Jacobian
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Assignment02Sol - Columbia University IEOR 4404: Simulation...

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