## Simulations

Here are links to some cool (and instructive!) java simulations. They are hosted at the Dartmouth Chance Project, an amazing resource, which is well worth checking out in its own right (in particular, the Chance News section).
• Central limit theorem demonstration. This is a java applet to demonstrate, in a quite dramatic fashion, the emergence of the normal distribution in sums of independent random variables. The applet allows you to specify the initial distribution by assigning probabilities for (up to) 10 values P(0), P(1), P(2), ..., P(9). The program then computes and plots, for given n, the distribution of a sum of n independent r.v.'s having this distribution. It is remarkable how quickly the distribution approaches the normal shape as one increases the value of n from n=1 (the original distribution) to 2, 3, etc. This occurs, sooner or later, no matter what the shape of the original distribution. For nice symmetric distributions (e.g., uniform) the convergence to the normal distribution is very fast; for other distributions it may take a bit longer to see the normal distribution emerge.
• Normal approximation to binomial distribution. A java applet that, for given values of n and p and a given range for k, plots the binomial distribution along with the approximating normal curve. The match is remarkably good, even if n is relatively small.
• Plot of the Binomial Distribution. This java applet plots the binomial distribution for a given value of p and three values of n.
• Coin tossing simulation. This applet simulates a sequence of coin tosses and plots the accumulated "scores" (1 for a head, -1 for a tail).