1) Classical MC: Estimate the parameter defined by: 2 = 2 1 dx (1+ x2 ) Using an Uniform distribution ~U(-2,2) in the interval [-2,2] with n=50: clear all n = 50; b=2; a=-2; LI=b-a; ThetaMC=0; x = rand(1, n); for I = 1 : n Sample = (2 * x(I) - 1) * 2; g = LI/(pi *(1 + Sample^2)); ThetaMC = ThetaMC + g/n; end 2) Antithetic variables Estimation of parameter from exercise 1 using antithetic variables clear all n = 50; ThetaAnt = 0; x = rand(1,n); for I = 1 : n SampleOrig = (2 * x(I) - 1)* 2; SampleAnti = (2 * (1 - x(I)) - 1)* 2; gOrig = 1/(pi * (1 + SampleOrig^2)); gAnti = 1/(pi * (1 + SampleAnti^2)); ThetaAnt = ThetaAnt + (gOrig + gAnti)/2*n; End 3) Control variables Estimate the parameter defined by: = 2 1 dx (1+ x2 ) From a sample of n=50. As control variate Y, it will be in interval [0,1/2] with an ~U(0,1). This is Y=I[0,1/2](U), being U~U(0,1) clear all n = 50; c* = 0.2956; v = rand(1, n); u = rand(1, n); for I = 1 : n ThetaC = 0; Mu = pi *(v(I) - 0.5); Sample = tan(Mu); if Sample >= 2 g = 1; else g = 0; end if u <= 1/2 Y = 1; else Y = 0; end ThetaC = ThetaC + g/n + 0.2956 * (Y - 1/2)/n; end Estimate the parameter defined by: = 2 1 dx 2 (1+ x ) 1) Using classical Monte Carlo method from a Cauchy distribution with parameter C(1,0) of sample size n = 16 . Calculate the variance of the estimator. Estimate the variance. 2) Using antithetic variables (using C(1,0), size n =16). Calculate the variance of the estimator. Estimate the variance. 3) Using control variates (using C(1,0), size n =16).Calculate the variance of the estimator. Estimate the variance. 4) What conclusions are derived from the above results