Probability Distribution Function and Shape

> restart;

> with(plots):

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The Negative Binomial Distribution

Consider observing independent Bernoulli trials having success probability p until r successes

have been observed. For example, flipping a coin until r heads are observed is

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

the first r successes then X is said to have the Negative Binomial ( r, p ) distribution.

If X is a random variable having the Negative Binomial ( r , p ) distribution, then the probability

distribution function for X is

f ( k ) = P( X = k ) = , k = r , r +1, r +2, ...

The form of this function can be seen by way of the following argument. The probability of any

fixed sequence of r successes and k-r failures has probability . How many such

sequences of r successes and k-r failures have a success on the k th trial? Namely,

binomial( k -1, r -1), as we are free to assign the first r -1 successes among the first k -1 trials.

The following code will draw a probability histogram for the Negative Binomial ( r , p ) distribution

for your choice of r and p .

> r:=4;

> p:=0.5;

> ProbHist(NegBinomialPDF(r,p,x),1..20,20);

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To see the effect of p on the shape of the Negative Binomial ( r , 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, assuming a fixed value of r .

> r:=4:

> for n from 0 to 9 do

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

> tracker[n]:=textplot([30,0.12,`p is `.num],color=blue):

> H[n]:=ProbHist(NegBinomialPDF(r,0.05+n*0.05,x),1..40,40):

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

> od:

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

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To see the effect of r on the shape of the Negative Binomial ( r , p ) distribution, the following

animation will draw a series of probability histograms as r varies from 1 to 20, assuming a

fixed value of p .

> p:=0.5:

> for r from 1 to 20 do

> num:=convert(evalf(r), string):

> tracker[r]:=textplot([60,0.25,`r is `.num],color=blue):

> H[r]:=ProbHist(NegBinomialPDF(r,p,x),1..80,80):

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

> od:

> display([seq(P[r], r=1..20)], insequence=true,title="p is fixed, r is increasing");

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