1 8. Let the integral I = = [ f(x) dx = 0 So COS (72) dx be evaluated using the Monte-Carlo method. 0 (a) Find the variance of the crude MCM estimator X = f(U), U~Unif (0, 1). (b) Suppose that a combination of stratified random sampling and an antithetic variate method of the form Y – = ½±(ƒ(U/2) + ƒ(1 − U/2)), U ~ Unif(0, 1), is applied to evaluate the integral. What is the efficiency gain of this estimator as compared to the crude Monte Carlo estimator? (c) How large a sample size do you need if you use the crude Monte Carlo of (a) or the antithetic stratified method of (b), respectively, in order to estimate the above integral, correct to three decimal places (i.e., the error does not exceed 10-3/2) with confidence of 95%? (d) Expand the function f(x) in Taylor's series and use the first two terms to construct a control variate. Don't analyze its variance nor compute the optimal parameter. The following trigonometric identities may be useful (or not): (5 – :) = sin (a) sin - = cos x and cos (b) cos² x = and sin² x = 1 + cos 2x 2 1 X - COS 2x 2 (c) sin x + cos x = √2 sin (1 + x)

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