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Statistical Data Analysis
Published in Timothy Bower, ®, 2023
The Poisson distribution is for random variables that count the number of events that might occur within a fixed unit, such as a quantity, a spatial region, or a time span. A special property of Poisson random variables is that their values do not have a fixed upper limit. Some examples might be the number of defects per 1,000 items produced, the number of customers that a bank teller helps per hour, or the number of telephone calls processed by a switching system per hour. It is denoted as X~Poi(λ). An example Poisson distribution probability mass plot is show in figure 3.3 for λ=3. Note that 0!=1, so k=0 is defined in the PMF.
Perception, Planning, and Scoping, Problem Formulation, and Hazard Identification
Published in Ted W. Simon, Environmental Risk Assessment, 2019
Given that no evidence exists that a swimmer has ever been affected by lightning while using an indoor pool, how should we calculate this risk? This example provides an introduction to the use of probability distributions. Here, we will consider the Poisson distribution, a collection of integer values that expresses the number of events occurring in a fixed interval given that the events occur independent of time. The Poisson distribution is defined by single parameter, the rate, or λ. The best-known historical application of the Poisson distribution is the estimate of the number of Prussian cavalry officers killed accidentally by horse kicks. These officers were uniformly superb horsemen, and accidental death by horse kick was a very rare event.
Reliability Statistics
Published in Ali J Jamnia, Khaled Atua, Executing Design for Reliability within the Product Life Cycle, 2019
Poisson distribution describes the probability of occurrence of certain numbers of an event in a continuous domain such as fixed time distance or length. It can be used in a manufacturing facility to model the expected number of defects in a given process. The main assumption is that the occurrences of these events are totally independent of each other, i.e., the occurrence of this event is random and has a constant rate. Also, events occur at certain intervals, and no two events happen at the same instant. The probability of a certain number of events in a fixed time interval is given by the following equation: Pkeventsinanintervalt=e−λλkk!
A novel mathematical optimization model for a preemptive multi-priority M/M/C queueing system of emergency department’s patients, a real case study in Iran
Published in IISE Transactions on Healthcare Systems Engineering, 2022
Erfaneh Ghanbari, Sogand Soghrati Ghasbe, Amir Aghsami, Fariborz Jolai
Upon patients’ arrival, provided an idle server, they will receive service, which takes an exponential time (Hu et al., 2018). According to Shortle et al. (2018), the exponential distribution is closely connected with the Poisson process. Therefore, we consider it from the perspective that the service process is a counting process in the form {N(t), t ≥ 0} where N (t) represents the number of people served up to the time t. Considering the notions mentioned for the Poisson process and knowing that if the number of events in a process follows a Poisson distribution, the time between two consecutive events will be exponential. Therefore, service time will be exponentially distributed. In addition, it can be said that according to the emergency discussion, the probability that this time will take a considerable amount is low, and the probability that it will take a small amount is high, which is entirely consistent with the exponential distribution.
Anchorage capacity reliability and redundancy optimization research in coastal ports
Published in Engineering Optimization, 2021
Zijian Guo, Yunzhuo Xu, Yong Yu, Zhijun Wei, Tianhan Xue, Wenyuan Wang, Ying Jiang
According to the existing studies on the distribution of interarrival time between two successive ships and the distribution of berth service time, the queuing theory model for arriving ships in port mainly includes several classical forms of M/M/S, M/Ek/S, Ek/Ek’/S, etc. (Song and Wang 2019; Tang et al.2016; Tang, Wang, et al.2014). In this study, the M/M/S queuing theory model is adopted initially as a representative by considering its common usage property in many existing port stochastic service systems (Tang, Wang, et al.2014; Tang, Guo, et al.2014; Hess, Kos, and Hess 2007). In this context, the ship arrival processes follow a Poisson distribution and the berth service time follows an exponential distribution. Application of other forms of queuing theory models could also be investigated by adopting different assumptions and can be considered a future research topic. The M/M/S queuing theory model is employed here, and Equations (10) and (11) are used to determine .
Design of Experiments for Generalized Linear Models
Published in Journal of Quality Technology, 2021
From Chapter 4 to Chapter 6, designs for different response distributions are discussed. They include binomial distribution, Poisson distribution, gamma distribution, etc. The binomial distribution is often invoked when we have binary response variables; while the Poisson distribution is useful for the counting type of response variables. The structures of these chapters are similar too. First, the specific GLMs and their information matrices are introduced. Then, optimal designs are derived in order for single predictor models, two predictor models, and other more complex models. Finally, how to obtain an exact design and the small sample size problem are discussed. One needs a good R coding skill to read and understand these chapters because many R codes are embedded in texts and they are not necessarily explained line-by-line. Also, it would be very helpful to comprehending these chapters if one runs those codes at the same time of reading.