(a,b,0) class of distributions
Encyclopedia
In probability theory
, the distribution of a discrete random variable
is said to be a member of the (a, b, 0) class of distributions if its probability mass function
obeys
where
(provided 
and 
exist and are real).
There are only three discrete distributions that satisfy the full form of this relationship: the Poisson
, binomial and negative binomial
distributions. 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. However, some further well-known distributions are available if the recursion above need only hold for a restricted range of values of k: for example the logarithmic distribution and the discrete uniform distribution.
The (a, b, 0) class of distributions has important applications in actuarial science
in the context of loss models.
and the negative binomial distribution
belong to this class of distributions, with each distribution being represented by a different sign of a. The more usual parameters of these distributions are determined by both a and b. The properties of these distributions in relation to the present class of distributions are summarised in the following table. Note that
denotes the probability generating function.
By multiplying both sides of the recursive formula by
, you get
which shows that the left side is obviously a linear function of
. When using a sample of 
data, an approximation of the 
's need to be done. If 
represents the number of observations having the value 
, then 
is an unbiased estimator of the true 
.
Therefore, if a linear trend is seen, then it can be assumed that the data is taken from an (a,b,0) distribution. Moreover, the slope
of the function would be the parameter
, while the ordinate at the origin would be 
.
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, the distribution of a discrete random variable
is said to be a member of the (a, b, 0) class of distributions if its probability mass functionProbability mass function
In probability theory and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value...
obeys
where
(provided and exist and are real).There are only three discrete distributions that satisfy the full form of this relationship: the Poisson
Poisson distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since...
, binomial and negative binomial
Negative binomial distribution
In probability theory and statistics, the negative binomial distribution is a discrete probability distribution of the number of successes in a sequence of Bernoulli trials before a specified number of failures occur...
distributions. 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. However, some further well-known distributions are available if the recursion above need only hold for a restricted range of values of k: for example the logarithmic distribution and the discrete uniform distribution.
The (a, b, 0) class of distributions has important applications in actuarial science
Actuarial science
Actuarial science is the discipline that applies mathematical and statistical methods to assess risk in the insurance and finance industries. Actuaries are professionals who are qualified in this field through education and experience...
in the context of loss models.
Properties
Sundt proved that only the binomial distribution, the Poisson distributionPoisson distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since...
and the negative binomial distribution
Negative binomial distribution
In probability theory and statistics, the negative binomial distribution is a discrete probability distribution of the number of successes in a sequence of Bernoulli trials before a specified number of failures occur...
belong to this class of distributions, with each distribution being represented by a different sign of a. The more usual parameters of these distributions are determined by both a and b. The properties of these distributions in relation to the present class of distributions are summarised in the following table. Note that
denotes the probability generating function.| Distribution |  |
 |
 |
 |
 |
 |
 |
|---|---|---|---|---|---|---|---|
| Binomial |  |
 |
 |
 |
 |
 |
 |
| Poisson Poisson distribution In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since... |
 |
 |
 |
 |
 |
 |
 |
| Negative binomial Negative binomial distribution In probability theory and statistics, the negative binomial distribution is a discrete probability distribution of the number of successes in a sequence of Bernoulli trials before a specified number of failures occur... |
 |
 |
 |
 |
 |
 |
 |
Plotting
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
. When using a sample of data, an approximation of the 's need to be done. If represents the number of observations having the value , then is an unbiased estimator of the true .Therefore, if a linear trend is seen, then it can be assumed that the data is taken from an (a,b,0) distribution. Moreover, the slope
Slope
In mathematics, the slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline....
of the function would be the parameter
, while the ordinate at the origin would be .