Some observations on the Monte Carlo simulations

Setting aside the triviality of the premise for “calculating” pi (i.e., one needs to be cognizant of the fact that the area of the circle is represented using a pi term in it, implying that pi is already known to the user), this simple example illustrates the basics of the Monte Carlo technique. The following general observations can be made from the pi-experiment:

•          A mathematical model needs to be in place in order to perform a sensitivity analysis of the system.
•          Given a model, a prediction can be made for the steady state output of the system, for a set of input variables.
•         Should a question arise on the validity of the outputs, the MC technique can be consulted. Specifically, the question can arise in the form of what variation can be statistically forecast in the outputs in the event of a variation in the inputs.
•          Having posed the question thusly, the parametric statistical variation can be performed to the inputs over a bounded interval, and the behavior of the mathematical model can be obtained.

If the system being analyzed is linear, a bell-curve distribution can be expected for large sample sizes randomized multiple times. One needs to be cognizant of the fact that MC simulations are highly statistical, and depend heavily on the validity of both the input variables and the mathematical model. In the case of the pi-estimation experiment, a sample size of 1000 can only have a discrete number of marbles fall within the confines of the circle. Thus, the accuracy of the estimate for pi is limited to the 1000^ths place; i.e., only three decimal places can be predicted for pi with a 1000-sample bin. For a 10,000 size bin, only a 4-digit accuracy can be predicted. However, repeated throws and subsequent averaging using the MC simulation can statistically result in values of pi that include far more number of digits. This is clear from the histograms shown earlier.

That greater accuracies can be seemingly predicted with a MC technique that is initially not available in the model is a rather disturbing thought. Adequate understanding of the Physics of the system being modeled, combined with proper bin sizes for the data analysis in a real-world system is thus an absolute necessity before one can launch off on an MC simulation to predict meaningful results.