Variance Reduction Using the Antithetic Variables Scheme

Most monte carlo estimates are based on iid samples, but algorithms based on negatively correlated samples maybe superior in terms of producing estimates with smaller variance.Suppose, we use monte carlo method to approximate mean, second moment, median and 75% quantile of a distribution with corresponding denstiy function: 2*X^3 I(x>1). The inverse method is usually used to generate samples follows such distribution. However, if we use both 1-U1 and U1 instead of U1. Here is R program to present variance reduction using both 1-U1 and U1.f(x)=2*X^3I(x>1)

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