### The binomial distribution

Introduction

The binomial distribution is the result of a Bernoulli process. In such a process each trial has only two outcomes and the result of each trial is independent of all previous trials. Examples of a Bernoulli process are the flips of a coin, the random walk of a drunken sailor in one dimension, and a system of noninteracting magnetic moments with spin 1/2.

To be specific we consider a system of N noninteracting magnetic moments (spins). Each spin has a probability p of being up and a probability q = 1 - p of pointing down. What is the probability that n spins are up? The answer is given by the binomial distribution:

PN(n) is the probability that n spins are up out of N spins.

The Program

Because PN(n) involves factorials, it is not easy to evaluate PN for large N. To handle N > 20, the program uses Stirling's approximation:

Questions

1. Calculate the form of PN(n) for N = 3 and arbitrary p by enumerating the eight possible microstates (configurations) and compare your result to the results of the program. (You can obtain the numerical results by selecting `Data Table` under the `Views` menu.)
2. Plot PN(n) for increasing values of N with p = 1/2. What is the qualitative dependence of the width of PN(n) on N? Also compare the relative heights of the maximum of PN(n).
3. The program plots PN(n) for various values of N in the same size window. Does the width of the distribution appear to become larger or smaller as N is increased?
4. Compare the plots of PN(n) versus n to the plots of PN(n) versus n/<n>. What can you conclude about the N-dependence of the width of PN?
5. Plot ln PN(n) versus n for N = 16. (Choose `Log Axes` under the `Views` menu.) Describe the behavior of ln PN(n). Can it be fitted to a parabola?

Java Classes

- BinomialApp

Updated 4 March 2009.