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:

P_{N}(n) is the probability that n spins are up out of N spins.

The Program

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

Questions

- Calculate the form of P
_{N}(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.) - Plot P
_{N}(n) for increasing values of N with p = 1/2. What is the qualitative dependence of the width of P_{N}(n) on N? Also compare the relative heights of the maximum of P_{N}(n). - The program plots P
_{N}(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? - Compare the plots of P
_{N}(n) versus n to the plots of P_{N}(n) versus n/<n>. What can you conclude about the N-dependence of the width of P_{N}? - Plot ln P
_{N}(n) versus n for N = 16. (Choose`Log Axes`under the`Views`menu.) Describe the behavior of ln P_{N}(n). Can it be fitted to a parabola?

Java Classes

- BinomialApp

Updated 4 March 2009.