07 June 2008

Virtual laboratories in probability and statistics

Virtual laboratories in probability and statistics offers several dozen applets for the visualization of probabilistic systems.

Most of them are pretty simple -- they graph some theoretical distribution and then show the results of repeating some corresponding experiment that gives that over and over again.

For example, the random walk experiment -- although I link to that one because it shows the arcsine distribution for a random walk quite well! The "arcsine distribution" as applied to random walks is as follows: let X1, X2, ... be independent random variables which are each +1 with probability 1/2 and -1 with probability 1/2. Let Sn = X1 + X2 + ... + Xn. Consider the sequence (S1, ..., S2n) -- now, what's the probability that the last zero in this sequence comes between time 2an and 2bn? It turns out that as n approaches infinity, this probability approaches
\int_a^b {1 \over \pi} {1 \over \sqrt{x(1-x)}} \, dx = {2 \over \pi} \left( \sin^{-1} \sqrt{b} - \sin^{-1} \sqrt{a} \right)

and in particular, the last zero is surprisingly likely to be near 0 or 2n, and surprisingly unlikely to be somewhere in the middle.

If you like pretty pictures (and you should -- a picture is worth a thousand words and all that), check out the section on interacting particle systems, which includes the fire process (which models the spread of a fire in a forest, where the trees form a grid and fires only spread between adjacent trees in discrete time -- so a bit unrealistic as a model for real fires, but who cares?) and the voter process. Here h particles in a grid change state in order to match their neighbors; "segregation" of the different particle types occurs, not because anybody's enforcing it from on high but just because of these local constraints, and some particle types just die out completely.

The web page not only includes the applets, but explanations of what's going on, since the intended audience here seems to be students taking a first course in probability.

1 comment:

Anonymous said...

I just came across something not really math related, but I thought it could be appreciated here.


If only every scientist/statistician could have such visualization tools!