# What are the pros and cons of learning about a distribution algorithmically (simulations) versus mathematically?

What are the pros and cons of learning about a distribution's properties algorithmically (via computer simulations) versus mathematically?

It seems like computer simulations can be an alternative learning method, especially for those new students who do not feel strong in calculus.

Also it seems that coding simulations can offer an earlier and more intuitive grasp of the concept of a distribution.

• the major con of the mathematical approach is to know the "corner" cases of the distribution. All the sample moments of any distribution exist, yet the distribution can have none such as Cauchy. In general both approaches should be combined. – mpiktas May 17 '11 at 18:36
• @mpiktas, I believe that you mean that the major pro is to know the corner cases :-). – NRH May 17 '11 at 20:47
• @NRH, yes, yes. Some neuron misfired probably :) – mpiktas May 18 '11 at 4:57

As a general remark, I find that one of the really important points to investigate with simulations is how distributions transform. In particular, in relation to test statistics. It is quite a challenge to understand that this single number you computed, the $t$-test statistic, say, from your entire data set has anything to do with a distribution. Even if you understand the math quite well. As a curious side effect of having to deal with multiple testing for microarray data, it has actually become much easier to show the students how the distribution of the test statistic pops up in real life situations.