Consider the integral I = f f e(x+y) dxdy . a) Estimate the integral I using...

  

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Consider the integral I = f f e(x+y) dxdy . a) Estimate the integral I using a simple Monte Carlo Method. Find 95% confidence interval for the true value of the integral. b) Estimate the integral I using data generated for a) and applying antithetic variate approach. Find 95% confidence interval for the true value of the integral. c) Estimate the integral I using data generated for a) and applying control variate approach. Use as control variate i) Y = U+V and ii) Y = (U + V). Note that you'll need to calculate exact (theoretical values) of E(Y) and E(Y2). Find 95% confidence interval for the true value of the integral. d) Compare a), b), c). Which variance reduction technique works better here. Consider the integral I = f f e(x+y) dxdy . a) Estimate the integral I using a simple Monte Carlo Method. Find 95% confidence interval for the true value of the integral. b) Estimate the integral I using data generated for a) and applying antithetic variate approach. Find 95% confidence interval for the true value of the integral. c) Estimate the integral I using data generated for a) and applying control variate approach. Use as control variate i) Y = U+V and ii) Y = (U + V). Note that you'll need to calculate exact (theoretical values) of E(Y) and E(Y2). Find 95% confidence interval for the true value of the integral. d) Compare a), b), c). Which variance reduction technique works better here.