BUILT-IN MATHEMATICA SYMBOL

NegativeMultinomialDistribution

represents a negative multinomial distribution with parameter n and failure probability vector p.

MORE INFORMATION

The probability for a vector of non-negative integers , , ..., , where is the length of in a negative multinomial distribution, is proportional to .

The parameter n can be any positive real number, and p can be any vector of non-negative real numbers that sum to less than unity.

If n is a positive integer, NegativeMultinomialDistribution gives the distribution of the failure counts in a sequence of trials with success probability 1-Total[p] and Length[p] types of failure before n successes occur.

NegativeMultinomialDistribution can be used with such functions as Mean, CDF, and RandomVariate.

EXAMPLES

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Basic Examples (4)

Probability density function:

Cumulative distribution function:

Mean and variance:

Covariance:

Probability density function:

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Cumulative distribution function:

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Mean and variance:

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Covariance:

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Scope (8)

Generate a set of pseudorandom vectors that follow a negative multinomial distribution:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Skewness:

Kurtosis:

Correlation:

Different mixed moments for a bivariate negative multinomial distribution:

Mixed central moments:

Mixed factorial moments:

Closed form for a symbolic order:

Mixed cumulants:

Hazard function:

Marginals do not simplify to known distributions:

Applications (1)

Find the distribution for the number of times a biased coin should be flipped until you get heads twice in a row. If you let p be the probability of heads, the two events for which you start over are

(tail) or

(head, tail) and the event for which you succeed is

(head, head). These events have the following probabilities:

The total number of coin flips until two heads:

Find the probability that no more than five coin flips are required:

Properties & Relations (3)

The components are correlated:

Relationships to other distributions:

A univariate negative multinomial distribution is a negative binomial distribution:

Possible Issues (3)

NegativeMultinomialDistribution is not defined when n is not a positive integer:

NegativeMultinomialDistribution is not defined when p is not a vector of probabilities that sum to less than 1:

Substitution of invalid parameters into symbolic outputs gives results that are not meaningful: