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MonteCarlo

# MonteCarlo - Monte Carlo Estimation and Simulation James...

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Monte Carlo Estimation and Simulation James Keesling In earlier sections of the course we have presented several ways to estimate integrals. These include, Newton-Cotes Closed and Newton-Cotes Open Formulas , Gaussian Quadra- ture , and Romberg Integration . In these methods we used exact algorithms and could cal- culate the error in the estimate in a precise way. The method we now introduce involves the random choice of points and a statistical estimate of the error. 1 Using Random Numbers to Estimate Integrals To start, we assume that we have a method for choosing points randomly from the unit interval so that the points come from a probability density function that is constant one on the interval. For practical purposes, the command rand () in your TI-89 will suffice. Suppose that we wish to estimate the integral R b a f ( x ) dx . The Monte Carlo Method does this by first choosing n random points from the unit interval [0 , 1]. Call this set of points { x i } n i =1 . Our formula for the estimate is given by the following calculation. b - a n · n X i =1 f ( x i ) Think of the formula as calculating the average value of the function f ( x ) and multi- plying this value by the length of the interval [ a, b ]. The average value for the function is determined by a random sample of the points in the interval [ a, b ]. The term Monte Carlo

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MonteCarlo - Monte Carlo Estimation and Simulation James...

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