Applications of Poisson Random Variables
Student project for ECE302
by Trevor Holloway
Contents
Poisson Random Variables
In 1837, the Poisson Distribution was introduced by Siméon Denis Poisson[1]. It has the pmf
$ P(X=x)= \frac{\lambda^x e^{-\lambda}}{x!} $
This distribution models the probablility that a number of events, x, will occur within a given time period when the average rate of occurance of such events on the same time interval is $ \lambda $.
Applications of Poisson Random Variables
Poisson random variables have many applications. This arises from the fact that many events in nature can be modeled as Poisson processes. Following are some examples of modern applications of the Poisson random variable.
Optimization
Poisson random variables are often used to model scenarios used to generate cost functions in optimization problems. For example, the economic lot scheduling problem aims to optimize the production of a certain number of products on a certain number of machines given a certain demand. If this is handled as a deterministic problem, it has been shown to be np-hard by Hsu in his 1983 paper on the topic [2]. Löhndorf & Minner have used Poisson random variables to make the problem stochastic, but also more feasible [3]. It should be noted that Löhndorf & Minner used a stuttering Poisson process to model their problem. Stuttering Poisson proceses are more generalized cases of Poisson processes wherein events occur at time periods dictated by a Poisson process but the number of events occurring within these time periods followas a geometric distribution [4].
Validation of Theoretical Research Models
Poisson random variables are used frequently in studies and research to model collected data. Such a strategy was proposed by researchers Iben Axén, et. al. on BioMed Central [5]. This article proposed the use of Poisson regression to find trends in data about lower back pain. The trendlines could then be compared to theoretical models to validate these models.
Poisson regression is a linear regression model that takes the form
$ \log (\operatorname{E}(Y|x))=\beta x $
Here, x is the data being fitted, Y is the resulting vector of best fit values, and beta is a regression parameter that must be solved for iteratively to complete the regression. This is done by iteratively solving the following equation:
$ \beta=(X^{T}WX)^{-1}X^{T}Wz $
Here, W is a diagonal matrix of all the mean values:
$ W=diag(\operatorname{E}(Y|x)) $
The elements of the vector z are given by the equation below:
$ z_{i}=\eta _{i}+(y_{i}-\mu _{i})/\mu _{i} $
It can be seen that Poisson random variables have many different applications in analysis, research, and optimization. These are simply a few of the potential applications.
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References:
[1]: Jerzy Letkowski. Applications of the Poisson probability distribution. 2012. Retrieved from http://www.aabri.com/SA12Manuscripts/SA12083.pdf
[2]: Wen-Lian Hsu. On the General Feasibility Test of Scheduling Lot Sizes for Several Products on One Machine. Management Science: Vol. 29, No. 1, Jan 1983. Retrieved from http://www.jstor.org/stable/2631168
[3]: Löhndorf & Minner. Simulation optimization for the stochastic economic lot scheduling problem. IIE Transactions: Volume 45, Issue 7, 2013. Retrieved from http://www.tandfonline.com/doi/full/10.1080/0740817X.2012.662310#tabModule
[4]: R. M. Adelson. Compound Poisson Distributions. OR: Vol. 17, No. 1, Mar 1966. Retrieved from http://www.jstor.org/stable/3007241
[5]: Iben Axén, et. Al. Analyzing repeated data collected by mobile phones and frequent text messages. An example of Low back pain measured weekly for 18 weeks. BMC Medical Research Methodology: July 2012. Retrieved from http://www.biomedcentral.com/1471-2288/12/105/
[6]: Joseph L. Schafer. Penn State University: 2006. Retrieved from http://sites.stat.psu.edu/~jls/stat544/lectures/lec15.pdf
