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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

  • 1
    Probably clearer if you state that $x$ is a random variable with a particular distribution up front, rather than asking us to compute $E(\sqrt[3]{x})$ before we even know that $x$ is random.2011-12-09
  • 0
    @ThomasAndrews ok thanks for your observation, don't kill me.... it wasn't so confusing, everyone could understand what i was meant. I didn't ask you to compute anything, I have already done that via algorithm.2011-12-09
  • 0
    Yeah, I wasn't meaning to attack, I just found reading the beginning of your problem difficult because the first sentence defines what looks like a function, and only late in the sentence do I understand the context. Just a writing note, really. You might start your problem with: I am trying to estimate $E(\sqrt[3]{x})$ where $x$ is a random real variable of Weibull distribution with parameters $(1,5)$. Defining $g$ causes confusion both because you don't use $g$ later in the sentence and because it makes $x$ look like a regular variable rather than a random variable.2011-12-09
  • 2
    Maybe this question would get more attention if it was on [CrossValidated](http://stats.stackexchange.com/).2011-12-10

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