I'm writing a random number generator test. The kit it will run on is way less powerful than a desktop PC, and I've come across this wiki article for determining $\pi$. I'm interested in the random walk method, utilising this formula:
There is also this explanatory graph with an example 5 walks of 200 steps each:
So I perform some number (n) of random walks of length L. If the generator is working correctly, I will calculate Pi to some accuracy approaching the true value. In this example, I will have used 5 * 200 = 1000 random numbers. So let's say that nL = 1000. My test will probably use about 1,000,000 random numbers.
The question is: what is the optimum value of n and L so that nL = 1,000,000 to get the best accuracy for calculating Pi? This will directly correlate to the quality of the random numbers.
That formula on Wikipedia is pretty misleading. That is not a limit, in fact, it is an equation: $$\pi={2n\over (E|W_n|)^2}={2\over (E|Z|)^2},$$ where $Z$ is a standard normal random variable. So your estimator for $\pi$ will simply be $$\widehat\pi={2\over \left({1\over n}\sum_{j=1}^n |Z_j|\right)^2},$$ where $(Z_j)_{j=1}^n$ is a random sample of standard normal values.