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The Negative Binomial Distribution
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The Negative NegativeBinomial Distribution
The negative binomial 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 is related to the binomial distribution. Where the
binomial distribution models the number of successful trials for a
given total number of trials, the negative binomial distribution
can be used to model the total number of trials for a given number
of successful trials.
The negative binomial distribution has two parameters: the
number of successful trials, and the probability of success in each
trial. If the number of trials is equal to 1, the negative binomial
distribution reduces to the geometric distribution.
Note that the distribution as defined here models only the
number of failures, not the total number of trials.
Practical examples 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 is sometimes called the
Pascal distribution or the Polya distribution.
The negative binomial distribution is implemented by the
NegativeBinomialDistribution
class. It has one constructor which takes two parameters. The first
parameter 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:
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NegativeBinomialDistribution negativeBinomial = new NegativeBinomialDistribution(12, 0.35); |
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Dim negativeBinomial As NegativeBinomialDistribution = _
New NegativeBinomialDistribution(12, 0.35) |
The NegativeBinomialDistribution class has two
specific properties.
NumberOfTrials returns the number of successful
trials.
ProbabilityOfSuccess returns the probabiltiy of
success of a trial.
NegativeBinomialDistribution has one static
(Shared in Visual Basic) method,
GetRandomVariate, which generates a random variate
using a user-supplied uniform random number generator.
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MersenneTwister random = new MersenneTwister();
double variate = NegativeBinomialDistribution.GetRandomVariate(random, 12, 0.35); |
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Dim random As MersenneTwister = New MersenneTwister()
Dim variate As Double = NegativeBinomialDistribution.GetRandomVariate(random, 12, 0.35) |
The above example uses the Mersenne
Twister to generate uniform random numbers.
For details of the properties and methods common to all discrete
probability distribution classes, see the topic on DiscreteDistribution
class.
Up: Discrete Probability Distributions Next: The Poisson Distribution Previous: The Hypergeometric Distribution Contents
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