Further notes about simulations

  

By now you should have written some simple simulation programs, and printed out the results. It is perhaps a good moment to pause and consider what are the aspects of simulations that we are interested in.

In general, we would like to get answers to questions that we don't know how to answer in any other way. But before we do that, we should develop some intuition on the cases that can check the answers. Therefore we begin with simulation of probabilities or averages that are known.

A

Simulation of probabilities/ averages that are known should address the following questions.

1. How close the simulation answers are to the theoretical answers? Print them side-by-side.
2. How large the simulation should be? Is it worth to change simulation size from 1,000 to 10,000 trials? In order to answer this question, your simulation has to provide ``relative" rather than absolute answers. (Answers of the form ``got 32 heads" are meaningless as they depend on simulation size!)
3. How do the answers change as we change the parameters? If you did a simulation of the fair coin, you could change the probability p from the usual value .

B

The next natural step is to extend models that we know how to handle both theoretically and by simulations to cover aspects that aren't easily accessible by theory. The sample questions involve

1. How would the answers change, if we allow perhaps more realistic assumptions in the model? As an example, suppose that we would like to model the birthday problem with people born non-uniformly throughout the year. Which way would you expect the answer to change?
2. What are typical errors of a simulation of size n? How can we estimate the accuracy of the answer without having the exact answer to compare it to? Chapter gives theoretical basis for such estimates.