Negative Binomial Distribution

The negative binomial distribution, also known as the Pascal distribution or the Polya distribution, models the number of failures before a specified number of successes in a series of Bernoulli trials. A Bernoulli trial is an experiment with two possible outcomes, labeled 'success' and 'failure,' where the probability of success has a fixed value for all trials.

The negative binomial distribution has two parameters: the number of successful trials $r$ and the probability of success $p$ in each trial. The probability mass function (PMF) is given by:

$$P(X=k)=(\genfrac{}{}{0ex}{}{k+r-1}{k})(1-p{)}^k{p}^r$$

The cumulative distribution function (CDF) is:

$$F(k;r,p)=\sum _{i=0}^k(\genfrac{}{}{0ex}{}{i+r-1}{i})(1-p{)}^i{p}^r$$

The domain of the negative binomial distribution is $k\in \{0,1,2,\dots \}$. The parameters must satisfy $r>0$ and $0<p\le 1$.

The negative binomial distribution is widely used in various fields due to its ability to model the number of failures before a specified number of successes. Common applications include:

• In jury selection, the number of rejected candidates before 12 jurors have been selected has a negative binomial distribution.

• When playing a video game where the probability of completing a level is constant, the total number of levels completed before the three lives are used up has a negative binomial distribution.

The negative binomial distribution has several important statistical properties:

The negative binomial distribution is closely related to several other distributions:

• The Binomial Distribution models the number of successes in a fixed number of Bernoulli trials.

• The Geometric Distribution is a special case of the negative binomial distribution with $r=1$.

The negative binomial distribution is implemented by the NegativeBinomialDistribution class. It has one constructor which takes two arguments. The first argument is the number of successful trials. The second parameter is the probability of success of a trial. The probability must be between 0 and 1. The following constructs a negative binomial distribution for 12 successes and probability of success 0.35:

var negativeBinomial = new NegativeBinomialDistribution(12, 0.35);

Dim negativeBinomial = New NegativeBinomialDistribution(12, 0.35)

No code example is currently available or this language may not be supported.

let negativeBinomial = NegativeBinomialDistribution(12, 0.35)

The NegativeBinomialDistribution class has two specific properties: NumberOfTrials returns the number of successful trials. ProbabilityOfSuccess returns the probability of success of a trial.

NegativeBinomialDistribution has one static (Shared in Visual Basic) method, Sample, which generates a random sample using a user-supplied uniform random number generator.

var random = new Pcg32();
int sample = NegativeBinomialDistribution.Sample(random, 12, 0.35);

Dim random = New Pcg32()
Dim sample = NegativeBinomialDistribution.Sample(random, 12, 0.35)

No code example is currently available or this language may not be supported.

let random = Pcg32()
let sample = NegativeBinomialDistribution.Sample(random, 12, 0.35)

The above example uses the Pcg32 class to generate uniform random numbers.

For details of the properties and methods common to all discrete probability distribution classes, see the topic on Discrete Probability Distributions.

• "Introduction to Probability Models" by Sheldon M. Ross.

• "Probability and Statistics" by Morris H. DeGroot and Mark J. Schervish.

Other Resources