This is just a quick note to demonstrate basic properties of the binomial distribution using Mathematica.

    First, initialize the enhanced plotting package:

(On some machines, this command might be just " <<Graphics")

Mathematica calls the "Choose" ratio of factorials "Binomial":

Name a list 'v' and fill it with integers to 20:

'Binomial' can operate on the list to compute all 21 values; let's call that list `bv':

BarChart can make a histogram of this distribution:

We could also use old reliable ListPlot instead:

In either case we see the characteristic 'bell curve' shape. Now let's normalize it to make the binomial distribution for N=20:

`%' means the previous result in Mathematica ; the above line takes the previous result and causes it to be evailuated Numerically:

Let's look at this normalized form:

We understand its structure much better by using a semi-log plot, which shows that the log of this distribution is very nearly parabolic, that is, the distribution is well-approximated by a Gaussian form (and hence justifies expanding the log of the distribution in our Taylor's series):

Left as an exercise to the student: Extend this to the general binomial distribution where p is not 1/2. Look at various limits, for example, the Poisson limit where p is `small' but N is `large'.