Supposing $g(x)=\sqrt[3]{x}$, I want to calculate the expected value of g, $E(\sqrt[3]{x})$, using Monte Carlo method, by generating $x_i$ from a Weibull distribution with parameters $(1,5)$.
After that, I want to use the control variates method and the antithetic method in order to to reduce the variance of my estimator, which I found with the simple Monte Carlo. And here is my problem, I do not know how to do these methods.
I would appreciate if someone could help me do that or give any tip/help.Thank you very much for your concern, in advance.
What I have done so far
Supposing $S$ is our estimator, then we know that $S=(\sum \limits_{i=1}^{N} g(x_i))/N$.
Using Matlab, I found the expected value $S$ by generating 1000 random numbers from the weibull(1,5)
distribution and calculate the sum. Here is my algorithm:
N=1000
sum=0;
for i=1:N;
X = wblrnd(1,5);
res(i)=X.^(1/3);
sum=sum+res(i);
end
S=sum/N