Triangulating Pi in Monte Carlo

I have made several posts on the basics of Monte Carlo integration techniques. I introduced the topic using MC integration to estimate the value of pi. Then I took a look at the way the convergence characteristics of the estimate depend upon the number of trials. Finally, I looked at how the relative size of the area being estimated affects convergence. Finally, I want to put some of these pieces together and introduce a final basic concept: the choice of the sampling distribution.

In Monte Carlo It Is Better To Be Full

This is a post in a series on the basics of Monte Carlo simulations. I introduced MC integration using it to estimate pi. Then I took a look at the way the convergence characteristics of the estimate depend upon the number of trials.

In my examples I was essentially using MC to do an integration: I have a search space and I sprinkle points over it and add up how many are under the curve. I know the size of the area over which I am sprinkling so I can use the ratio of points under the curve to total points to estimate the area under the curve. In this post, I want to address another aspect of convergence: what is the relationship between the accuracy of the integration and the fraction of total area taken up by the area of interest?

i.e. if the area of interest only occupies 25% of the total area, what difference does that make to the choice of the number of points to use compared to when the area of interest occupies, say 75%?