6.10 Use Monte Carlo integration with antithetic variables to estimate e Ꮖ dx, 1+x2 and find the approximate reduction in variance as a percentage of the variance without variance reduction.

Generate a random sample of standard lognormal data (rlnorm()) for sample size n = 100. Construct histogram estimates of density for this sample using Sturges’ Rule, Scott’s Normal Reference Rule, and the FD Rule.

using r language Obtain a bootstrap t confidence interval estimate for the correlation statistic in Example 8.2 (law data in bootstrap).

Compute a Monte Carlo estimate of 0.8 by sampling from Uniform(0,0.8) and estimate the variance of ⑦.

Write a function to compute a Monte Carlo estimate of the Gamma(r = 3, λ = 2) cdf, and use the function to estimate F(x) for x = 0.2, 0.4, . . . , 2.0. Compare the estimates with the values returned by the pgamma function in R.