Assignment02Sol

# Assignment02Sol - Columbia University IEOR 4404 Simulation...

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 3

Assignment02Sol - Columbia University IEOR 4404 Simulation...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online