< List of probability distributions

The Gamma Poisson distribution (GaP) is a two-parameter mixture model that combines the Gamma and Poisson distributions. It is used to model various data types, from failure rates to RNA-Sequencing data. In this blog post, we dive deeper into understanding what the Gamma Poisson distribution is and its applications.

## Gamma Poisson distribution properties

The Gamma distribution is used to model continuous random variables that are always positive, while the Poisson distribution models random events that occur in a given interval. When the Gamma distribution doesn’t fit data because the overall distribution from multiple samples is too spread out (i.e., the variance is greater than the mean), the Gamma Poisson model may be a good fit.

The probability mass function (PMF) of the gamma Poisson distribution is [1]

Where Γ is the gamma function.

The negative binomial distribution is closely related to the gamma Poisson distribution. This can happen when the probability density distribution across sample plots follows a gamma distribution with scale dependent parameters and a constant mean [2]. Thus, the GaP process is sometimes called the negative binomial process [3] although it is a *form *of the negative binomial rather than its equivalent [4].

## Use of the gamma Poisson distribution

The Gamma Poisson distribution is used to model various data types, including failure rates, which represent the probability of failure per unit of time. In a study conducted by researchers from the University of Waterloo, they analyzed failure rates of various components in vehicles and found that the Gamma Poisson distribution was a good fit for modeling these rates [5]. The Gamma Poisson can also be used to model RNA-Sequencing data, which measures the expression levels of genes. In a study published by Oxford University Press, researchers used the Gamma Poisson distribution to analyze gene expression data [6].

Another application of the Gamma Poisson distribution is modeling the random distribution of microorganisms in a batch of food. A group of European researchers found that the distribution of microorganisms in food followed a Gamma Poisson distribution [7]. This information is crucial for food safety as it helps in identifying the type and amount of micro-organisms in food.

The Gamma Poisson distribution can also be used in Bayesian statistics, where the prior distribution is a Gamma distribution, and the likelihood is a Poisson distribution. Bayesian inference with the Gamma Poisson distribution has been used to analyze data in ecology, such as estimating the variation of plant populations [8].

In conclusion, the Gamma Poisson distribution is a powerful tool for modeling various data types with two positive parameters, α and β. It combines the Gamma and Poisson distributions and is particularly useful when the Gamma distribution doesn’t fit data due to a high variance. With its applications in modeling failure rates, RNA-Sequencing data, micro-organisms in food matrices, and Bayesian statistics, the Gamma Poisson distribution proves to be a versatile model that can be applied to various fields. As college students, understanding the Gamma Poisson distribution gives us an additional tool to model real-world data and make informed decisions based on our analysis.

## References

[1] Gamma-Poisson Distribution. Retrieved April 11, 2023 from: http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Gammapoisson.pdf

[2] Magnussen, S. (2008). A Gamma-Poisson Distribution of Point to k Nearest Event Distance. Forest Science 54(4): 429-441.

[3] Ewing, S. & Kunz, M. Fitting of Failure Rate Data to Gamma-Poisson Distribution Utilizing Method of Moments. Retrieved January 8, 2021 from: https://inldigitallibrary.inl.gov/sites/sti/sti/8059210.pdf

[4] Engelhardt, M. (1994). Events in Time: Basic Analysis of Poisson Data.

[5] Crowder, M. J., et al. “Analysis of Reliability and Warranty Claims in Products With Age and Usage Scales.” *Technometrics*, vol. 51, no. 1, 2009, pp. 14–24. *JSTOR*, http://www.jstor.org/stable/40586559. Accessed 11 Apr. 2023.

[6] Constantin Ahlmann-Eltze, Wolfgang Huber, glmGamPoi: fitting Gamma-Poisson generalized linear models on single cell count data, *Bioinformatics*, Volume 36, Issue 24, 15 December 2020, Pages 5701–5702

[7] Rob D Reinders, Rob De Jonge, Eric G Evers,

A statistical method to determine whether micro-organisms are randomly distributed in a food matrix, applied to coliforms and Escherichia coli O157 in minced beef, Food Microbiology, Volume 20, Issue 3, 2003, Pages 297-303,

ISSN 0740-0020, https://doi.org/10.1016/S0740-0020(02)00134-X.

[8] Engelhardt, M. (1994). Events in time: Basic analysis of Poisson Data. U.S. Dept. of Enery.