Originally Posted by
CaptainBlack I'm not entirely clear what you mean by prove in this case. As most Monte-Carlo methods will give as estimate of the integral rather than its exact value (here I exclude tricks like Martin Baal's martingale based variance reduction technique which for his toy problem gave the exact value of the pertinent integral).
What we can do is by a bit of manipulation show that:
$\displaystyle E(e^{-x^2})=\frac{k}{\sqrt{2\sigma^2+1}}$
when $\displaystyle x \sim N(0,\sigma^2)$, for some $\displaystyle k$ (a very specific $\displaystyle k$ given by the integral remaining after the jiggery-pokerry required to get all the $\displaystyle \sigma$ dependence outside the integral).
Then show that $\displaystyle k \approx 1$ by Monte-Carlo methods.
RonL
To demonstrate:
Code:
This is EULER, Version 2.3 RL-06.
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Enter command: (20971520 Bytes free.)
Processing configuration file.
Done.
>
>sigma=2; ..set a nominal sigma of 2
>
>x=normal(1,10000)*sigma;..genrt sample of X~N(0,sigma^2)
>ii=exp(-x^2); ..calculate exp(-x^2) for the sample of X
>
>..now estimate the expectation as the average:
>..============================================
>
>{expectation,sdexp}=mean(ii)
0.335965
>
>..estimate k as the expectation times sqrt(2*sigma^2+1)
>..=====================================================
>
>kk=expectation*sqrt(2*sigma^2+1)
1.0079
>
>..estimate the standard error of expectation:
>..-------------------------------------------
>
>SEexp=sdexp/sqrt(length(ii))
0.00335965
>
>..estimate the se of the estimate of kk:
>..--------------------------------------
>
>SEkk=SEexp*sqrt(2*sigma^2+1)
0.010079
>
RonL