Using Monte Carlo Integration and Control Variates to Estimate

We will demonstrate the utility of Monte Carlo integration by using this algorithm to calculate an estimate for . In order to improve this estimate, we will also demonstrate how a family of covariate functions can be used to reduce the variance. Finally, the optimal covariate function within this family is found numerically. 1. An Introduction to Monte Carlo Integration 1.1. Monte Carlo Integration. Monte Carlo Integration is a method for approximating integrals related to a family of stochastic processes referred to as Monte Carlo Simulations. This term was coined in the mid 1950’s by Nicholas Metropolis, one of the original scientists of the Manhattan Project, to reference the random processes of this class of algorithms. The method relies on the construction of a random sample of points so outputs are non-unique; however, these outputs probalistically converge to the actual value of the integral as the number of sample points is increased. Since its development, Monte Carlo Integration has been used to evaluate many problems which otherwise become computationally inefficient or unsolvable by other methods. 1.2. Algorithm. To evaluate I = ∫ b a f(x) dx by Monte Carlo Integration, first generate a sequence of N uniformly distributed random variables within the interval. That is, create Xi ∼ U [a, b] and let Yi = f(Xi) for 1 ≤ i ≤ N . Find the average ȲN and multiply this value by the length of the interval, (b − a), for an approximation of I. Of course, keep in mind that in general larger choices of N provide better approximations for I. An example of this process is tossing rocks into a circular pond for an estimation of . If we enclose a circular pond of radius r = 1 with a square having sides of length 2, we will see that Asquare ∗ n N ≈ where n is the number of rocks in the pond and N is the number of rocks within the square. Later, we propose another method to estimate with MCI. 1 2 N. CANNADY, P. FACIANE, AND D. MIKSA 1.3. Derivation. Definition 1.1. [1] For any continuous random variable X ∼ (X) and Y = f(X), the expected value of Y is defined as: E[Y ] = E[f(X)] = ∫ ∞