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By W. J. Padgett

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Accessing a peripheral several thousands or perhaps millions of times can slow down a simulation. Therefore, the preferred approach is to generate pseudo-random numbers at run-time, using a specified deterministic recurrence equation on integers. This allows fast generation, eliminates the storage problem, and gives a reproducible sequence. However, great care is needed in selecting an appropriate recurrence, to make the sequence appear random. Simulation and Monte Carlo: With applications in finance and MCMC © 2007 John Wiley & Sons, Ltd J.

The only technical problem is how to generate points uniformly over the region C. Usually, this is done by enclosing C within a rectangle D with sides parallel to the u and v axes. Generating points uniformly within D is easy. If such a point also falls in C it is accepted and V/U (a ratio of uniforms) is accepted; otherwise it is rejected. 4 Devise a ratio of uniforms algorithm for sampling from the density proportional to h x = exp −x on support 0 . Solution. 8) Note how C is the closed region bounded by the curves v = 0 v = −2u ln u.

The pseudo-code below gives a method for shuffling the order of numbers in a sequence, where the first five numbers are entered into the array T j j = 0 4 . Problems 35 Output T (4) into shuffled sequence N = 4T 4 T 4 =T N Input next U from unshuffled sequence T N =U Obtain the first six numbers in the shuffled sequence. 12. Consider the generator Xn+1 = 5Xn + 3 mod 16. Obtain the full cycle starting with X0 = 1. 3 with k = 6. Find the first 20 integers in the shuffled sequence. 13. 0 Frequency 1104 969 961 850 What are your conclusions?

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