# Stuttering Poisson Distribution

The Stuttering Poisson Distribution (SPD) is a non-negative discrete compound Poisson distribution that describes two or more events that happen in quickly in bursts. For example, the events might occur in groups or batches .

The distribution has the probability generating function (PGF):

Where Px is the probability a discrete random variable will take the value x.

## Calculating Stuttering Poisson Distribution Probability

A general formula for calculating the probability of observing a demand equal to x is given by :

For low demand (x = 1 or x = 2), the formula simplifies to

The following table shows Poisson (λ = 2) and stuttering Poisson distribution (λ = 1 and  ρ = 5) probabilities and cumulative probabilities:

The term “stuttering” Poisson is mostly used in older literature, it does make an appearance in a few modern texts. Many modern authors call the distribution a Poisson-stopped sum or multiple Poisson. The SPD does have a variety of other names in the literature. For example, Cox  called the process a “cumulative process associated with a Poisson process.” It’s also referred to as:

• Composed distribution 
• Compound Poisson ,
• Distribution par grappes ,
• Poison distributions with events in clusters 
• Poisson power-series distribution 
• Pollaczek-Geiringer distribution.

The name “stuttering” Poisson distribution originated with Galliher et al. .  Patel  introduced the triple- and quadruple stuttering Poisson distributions.

References

 Huiming, Z. et al. (2012). Some Properties of the Generalized Stuttering Poisson Distribution and Its Applications. Studies in Mathematical Sciences. Vol. 5, No. 1, 2012, pp. [11–26] www.cscanada.net DOI: 10.3968/j.sms.1923845220120501.Z0697

 Syntetos, A. & Boylan, J. (2021). Intermittent Demand Forecasting. Wiley.

 Cox, D. R. (1962). Renewal Theory, Methuen, London

 Janossy L. et al. (1950). 0n composed Poisson distributions, I, Acta. Math. Acad. ScL Hung., 1, pp. 209–224.

 Feller, W. (1957). An Introduction to Probability Theory and Its Applications (2nd ed.). Vol 1. New York. Wiley.

 Thyrion, P. (1960). Note sur les distribution “par grappes.” Association Royal des Actuaires Belges Bulletin, 60. 49-66.

 Castoldi, L. (1963). Poisson processes with events in clusters. Rendiconti del Seminaro della Facolta di Scienze della Universita di Cagliari, 33, 433-437.

 KHATRI, C. G. & PATEL, I. R. (1961). Three classes of univariate discrete distributions. Biometrics, 17, 567-75.  GALLIHER, H. P., MORSE, P. M. and SIMOND, M. (1959). ‘ Dynamics of Two Classes of Continuous-Review Inventory Systems ‘, Opns. Res. 7, 362-383.

 Patel, Y. C. (1976). Estimation of the parameters of the triple and quadruple stuttering Poisson distributions. Technometrics, 18,  67-73.

### One response to “Stuttering Poisson Distribution”

1. Joe says:

Nice!