In probability theory, a member of the (a, b, 0) class of distributions is any distribution of a discrete random variable N whose values are nonnegative integers whose probability mass function satisfies the recurrence formula for some real numbers a and b, where
Only the Poisson, binomial and negative binomial distributions satisfy the full form of this relationship.
These are also the three discrete distributions among the six members of the natural exponential family with quadratic variance functions (NEF–QVF).
More general distributions can be defined by fixing some initial values of pj and applying the recursion to define subsequent values.
This can be of use in fitting distributions to empirical data.
The (a, b, 0) class of distributions has important applications in actuarial science in the context of loss models.
The more usual parameters of these distributions are determined by both a and b.
denotes the probability generating function.
; it coincides with the negative binomial distribution for positive, finite real numbers
, and it equals the binomial distribution for negative integers
An easy way to quickly determine whether a given sample was taken from a distribution from the (a,b,0) class is by graphing the ratio of two consecutive observed data (multiplied by a constant) against the x-axis.
By multiplying both sides of the recursive formula by
, you get which shows that the left side is obviously a linear function of
represents the number of observations having the value
Therefore, if a linear trend is seen, then it can be assumed that the data is taken from an (a,b,0) distribution.