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Assuming I want to get an estimation of $\pi$ using a MonteCarlo estimation of the unit circle. The most natural choice is to take random points in the smallest domain containing the circle, such as $[-1,1]\times[-1,1]$. It seems to be that this is a better choice than taking $[-10,10]\times[-10,10]$ because it is more ``sensitive'' to the geometry I am integrating. I guess I can tie this to the variance of the estimator.

We have Bernouilli law with variance $$\frac{p(1-p)}{N}$$ so it seems that $p\sim 1$ does not minimize the variance.

What I am missing?

Learning is a mess
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1 Answers1

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The surface of the unit circle is $\frac{\pi}{4}$.

If you have a geometrical approach, and you choose a one-by-one square, the probability that your generator hits the unit circle is $\frac{\frac{\pi}{4}}{1*1}$. However, if you choose a bigger square, your probability to hit the circle becomes $\frac{\frac{\pi}{4}}{10*10}$.

You can get the same results by assuming X and Y are your uniform variables, and calculate the probability of the event $\{X^2+Y^2<1\}$, therefore , you would want to study the variance of $X^2+Y^2$.

Canardini
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