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Assignment02

Assignment02 - (a Estimate E N by generating 100 values of...

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Columbia University IEOR 4404: Simulation Fall 2009 Solution to Assignment 2, due on Sep. 24th 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 .] 2. Use simulation to approximate Cov( U,sin ( U )), where U is uniform on (0 , 1). Compare your approxi- mation with the exact answer. 3. For uniform (0 , 1) random variables U 1 ,U 2 ,... deﬁne N = Minimum ( n : n X i =1 U i > 1 ) That is, N is equal to the number of random numbers that must be summed to exceed 1.
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Unformatted text preview: (a) Estimate E [ N ] by generating 100 values of N . (b) Estimate E [ N ] by generating 1 , 000 values of N . (c) Estimate E [ N ] by generating 10 , 000 values of N . (d) What do you think is the value of E [ N ]? 4. Let U i ,i ≥ 1, be random numbers. Deﬁne N by N = Maximum ( n : n Y i =1 U i ≥ e-3 ) where Q i =1 U i ≡ 1. (a) Find E [ N ] by simulation. (b) Find P ( N = i ), for i = 0 , 1 , 2 , 3 , 4 , 5 , 6, by simulation....
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