This figure shows the sort of theory that I want you to create

Here's a graph of five estimates of π, as a function of R. Also two theoretical curves (in grey), showing a predicted interval, inside which we expect the estimate to lie (with high probability). We can use this sort of theory to decide how many points R we need in order to get a required accuracy.

I want you to make a theory that makes predictions like the grey lines above. You can do it!

This page includes the python program that made the coloured estimates of π (but not the theory curves).

The next figure shows another 5 random runs with the same theory curves.



David J.C. MacKay