Probability Distribution Function and Shape

> restart;

> with(plots):

>

The Geometric Distribution

Consider observing independent Bernoulli trials having success probability p until the first

success is observed. For example, flipping a coin until the first time a head is observed is

such an exercise. If we define a random variable X as the number of trials required to observe

the first success then X is said to have the Geometric( p ) distribution.

If X is a random variable having the Geometric( p ) distribution, then the probability distribution

function for X is

f ( k ) = P( X = k ) = , k = 1,2,3, ...

This is readily seen by noting that if Bernoulli trial number k yields the first success, then the first

k -1 trials were all failures.

The following code will draw a probability histogram for the Geometric( p ) distribution

for p= 0.5, but you are encouraged to try other values of p.

> p:=0.5;

> ProbHist(GeometricPDF(p,x),1..15,15);

>

To see the effect of p on the shape of the Geometric( p ) distribution, the following

animation will draw a series of probability histograms as p varies from 0.05 to 0.5 by

increments of 0.05

> for n from 0 to 9 do

> num:=convert(evalf(0.05+n*0.05), string):

> tracker[n]:=textplot([12,0.4,`p is `.num],color=blue):

> H[n]:=ProbHist(GeometricPDF(0.05+n*0.05,x),1..15,15):

> P[n]:=display({H[n],tracker[n]}):

> od:

> display(seq(P[n], n=0..9), insequence=true,title="p is increasing");

>