probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...

and statistics the extended negative binomial distribution is a

discrete probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...

extending the

negative binomial distribution In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expr ...

. It is a truncated version of the negative binomial distribution for which estimation methods have been studied.Shah S.M. (1971) "The displaced negative binomial distribution", ''Bulletin of the Calcutta Statistical Association'', 20, 143–152 In the context of actuarial science, the distribution appeared in its general form in a paper by K. Hess, A. Liewald and K.D. Schmidt when they characterized all distributions for which the extended

Panjer recursion The Panjer recursion is an algorithm to compute the probability distribution approximation of a compound random variable S = \sum_^N X_i\, where both N\, and X_i\, are random variables and of special types. In more general cases the distribution of ...

works. For the case , the distribution was already discussed by Willmot and put into a parametrized family with the

logarithmic distribution In probability and statistics, the logarithmic distribution (also known as the logarithmic series distribution or the log-series distribution) is a discrete probability distribution derived from the Maclaurin series expansion : -\ln(1-p) = p ...

and the negative binomial distribution by H.U. Gerber.

Probability mass function

For a natural number and real parameters , with and , the

probability mass function In probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. The probability mass ...

of the ExtNegBin(, , ) distribution is given by :

f(k;m,r,p)=0\qquad \textk\in\

and :

f(k;m,r,p) = \frac\quad\textk\in\textk\ge m,

where :

= \frac = (-1)^k\,\qquad\qquad(1)

is the (generalized)

binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...

and denotes the

gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except th ...

Probability generating function

Using that for is also a probability mass function, it follows that the

probability generating function In probability theory, the probability generating function of a discrete random variable is a power series representation (the generating function) of the probability mass function of the random variable. Probability generating functions are often ...

is given by :

\begin\varphi(s)&=\sum_^\infty f(k;m,r,p)s^k\\
&=\frac

\qquad\text , s, \le\frac1p.\end

For the important case , hence , this simplifies to :

\varphi(s)=\frac
\qquad\text, s, \le\frac1p.

References

{{ProbDistributions, discrete-infinite Discrete distributions Factorial and binomial topics