Negative Hypergeometric Distribution

Negative hypergeometric distribution, also known as the inverse hypergeometric or hypergeometric waiting-time distribution, is of interest in applications of inverse sampling without replacement from a finite population where a binary observation is made on each sampling unit. For example, imagine that a credit card company estimates that 18 of its 583 platinum card holders have gained their wealth illegally. A banking regulator is to perform an audit. “How many randomly selected platinum card holders will they have to investigate before finding a criminal?” It should be used in any situation where there is hypergeometric sampling and one is asking the question, “How many failures will I observe before I get s successes?” or alternatively “How many samples do I need to have s successes?” Sampling is performed by randomly choosing units sequentially one at a time until a specified number of one of the two types is selected for the sample. The probability distribution function for a random variable that has a negative hypergeometric distribution was described by Samuel Wilks (1963), discussed by Norman L. Johnson and Samuel Kotz (1969), and developed by William C. Guenther (1975). Daniel Zelterman (2005) presented some variations of the negative hypergeometric distribution.

Discrete distributions, such as the binomial, geometric, hypergeometric, and negative binomial, are discussed in most calculus-based introductory statistic books. But the negative hypergeometric has not appeared often in such books or in literature. Generally, many people have used all of these discrete distributions in a probability unit, often without even realizing it. That is, they have calculated probabilities from scratch using the ideas of permutations and combinations. For example, a person may be asked to find the probability when a fair coin is tossed six times that exactly two are heads. Even though this example follows a binomial distribution, the person learns how to construct this probability prior to ever hearing the name of that distribution. This entry aims to distinguish rather than confuse binomial, hypergeometric, negative binomial, and negative hypergeometric distributions and to understand the general form of the probability mass functions, the expected value and variance of the negative hypergeometric distribution.

Discrete distributions can be divided into the two categories as to sampling: from either an infinite- or a finite-sized population. Finite-sized populations are assumed to be sampled without replacement. In a finite population, the probability of future events depends on the composition of individuals already sampled. In infinite populations, the distribution of future events is independent of those of the past.

The binomial distribution describes the number of successes when a predetermined number of items are independently sampled from an infinitely large Bernoulli population. The hypergeometric distribution is the corresponding model when the population has a finite size. The negative binomial and negative hypergeometric distributions describe the number of trials necessary in order to obtain a specified number of “successes.” It describes properties of a discrete-valued random variable Y whose support is taken to be 0, 1,… or a finite subset of these nonnegative integers.

The simplest example of a random variable is the Bernoulli random variable for

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