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.

Write a function to compute a Monte Carlo estimate of the Beta(3, 3) cdf, and use the function to estimate F(x) for x = 0.1,0.2,...,0.9. Compare the estimates with the values returned by the pbeta function in R.

using r language

here is example 7.6## Example 7.6 Suppose the sample population is χ 2 (2), which is non-normal but with same variance 4. ▶ Repeat the simulation, but replacing the N(0, 4) samples with χ 2 (2) samples. ▶ Calculate the empirical confidence level.(Empirical confidence level) n <- 20 alpha <- 0.05 UCL <- replicate(1000, expr = { x <- rchisq(n,df=2) (n-1)*var(x)/qchisq(alpha,df=n-1) }) sum(UCL >4) mean(UCL > 4) ## t.test function n <- 20 x <- rnorm(n,mean=2) result <- t.test(x,mu=1) result$statistic result$parameter result$p.value result$conf.int result$estimate