MATH 375 -- Probability and Statistics 1

Binomial and Geometric Distributions

Week of September 22, 2003

>	read "/home/fac/little/public_html/ProbStat/MaplePackage/MSP.map";

1)  The binomial probability distribution function in our package is called

BinomialPDF.   For instance, with   n = 15 trials, and success probability

p = .2, the probability of 10 successes is

>	BinomialPDF(15,.2,10);

The probability of fewer than 5 successes is

>	add(BinomialPDF(15,.2,k),k=0..4);

Here is a plot of BinomialPDF(15,.2,k), for  k = 0 .. 15, ``connecting the dots'':

>	plot([seq([k,BinomialPDF(15,.2,k)],k=0..15)]);

For fixed   n ,  increasing   p  skews the binomial distribution to the right.

For instance, here are BinomialPDF(15,.5,k) and BinomialPDF(15,.8,k)

>	plot([seq([k,BinomialPDF(15,.5,k)],k=0..15)]);

>	plot([seq([k,BinomialPDF(15,.8,k)],k=0..15)]);

What happens if we fix p  and increase n?

>	plot([seq([k,BinomialPDF(16,.2,k)],k=0..16)]);

2) The geometric probability distribution function is called GeometricPDF

>	GeometricPDF(.3,6);

>	plot([seq([k,GeometricPDF(.2,k)],k=1..20)]);

>	plot([seq([k,GeometricPDF(.4,k)],k=1..20)]);

The larger   p  , the faster the geometric probability distribution ``dies off''

(recall the random variable Y here is the number of the trial on which

the first success is observed -- that happens sooner (on average), the

bigger   p   is.