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Questions tagged [poisson-process]

For questions about the theory or applications of the Poisson process, one of the most widely applied point processes in statistics and elsewhere.

The Poisson process is one of the most widely used point processes, as well as one of the most important objects of study in the theories of point processes and of (more general) stochastic processes. Its name is derived from the fact that, for a Poisson point process, the number of points in a region of finite size is a random variable with a Poisson distribution.

The Poisson process can be defined on many different types of spaces. The simplest definition occurs on the real line, where the distance between two consecutive points of the process will have an exponential distribution. This implies that the points have the memoryless property: intuitively, the process "restarts afresh" at every point.

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What are finite window effects?

I'm reading a paper that uses a Poisson process to model real world events. The authors mention "finite window effects". What are finite window effects? Here is quote from the paper where the authors first mention the term: If the data come from a…
slayton
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Exponentially decaying integral of a Poisson process

Suppose that $X_t$ is the set of times of the events of a Poisson process with unit rate after $t$ seconds. (In other words, $X_t$ is a set of $N$ uniformly distributed points over $[0,t]$ where $N$ is Poisson distributed with mean $t$.) Let $Y_t =…
Neil G
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Fit to a non-homogeneous Poisson process

I have a sequence of event timings, $t_1, t_2, t_3, ..., t_n$ where there are a total number of $n$ events happening. For example, they are the timings when 911 was called in a city. I know these events is a non-homogeneous (or inhomogeneous)…
Shawn Wang
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Poisson process thinning females and males arriving

Rock tickets are sold at a ticket counter. Females and males arrive at times of independent Poisson processes with rates 30 and 20. What is the probability that the first three customers are female My Work Let $F(t), M(t)$ be the number of…
Moderat
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Poisson process Continuous -time stochastic processes

Duronto Express arrives at the Bombay Central station according to a Poisson process of rate 3 trains/hour. Local Line trains arrive according to a Poisson process of rate 4 trains/hour. Conditionally on the event that 8 trains arrive from 9 am to…
bluelagoon
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Expected waiting time

The following is a worked example found in past papers of my university, but haven't been able to figure out to solve it (I have the answer, but do not understand how to get there). Any help in enlightening me would be much appreciated. A store…
avriis
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Confusion about Poisson rectangular pulse model

I am reading a paper by Rodriguez-Iturbe et al. from 1986 and am confused by the below derivation. The model they are working with is a Poisson process with rate $\lambda$ in which each occurrence in the process corresponds to a storm. These storms…
user272429
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Error prediction for poisson process

A Poisson process has rate of N per unit length per unit time. The events occur uniformly on the x-axis but I always have measurement error of greater than 1/N. I was wondering is it possible that I can use my past measurements to reduce the error…
user2555272
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Crossing a road through a Poisson process

Am currently working on a Stochastic Poisson process on my project. I have thought and settled on the below scenario which I think is appropriate. However, solving it am not getting what I expect. I want to cross a road at a spot where cars pass…
Ben
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Help with Poisson Process

I'm going to repost this here since my questions never get answered on mathstackexchange. It might be better suited to this location, as well. At the end of the workday, I add an amount between 0 and 1 dollar to my change container. The amount added…
user288742
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Is there a similarity between a jump process and a counting process since both follow a Poisson distribution

I read that a jump in a stock price can be modeled as a Poisson process. But I have also read that a Poisson process is a good model for a counting process (i.e. number of hits to a website per unit time). Intuitively, I am not able to see the link…
Victor
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How to use Poisson statistics to estimate error bars on rates

I want to know the fraction of bananas which are ripe at some moment in time. If I have 100 bananas, and 25 of them are ripe, then my rate is 25%. But if I want error bars on this so I can generalize – would I use N=100 and say "25 ± 10 bananas are…
tomr_stargazer
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How to implement the Kolmogorov forward equation for a Poisson process?

How could I implement the Kolmogorov forward equation on the Poisson process? I know that Kolmogorov forward equation is: $$ P_{ij}'=\sum_{k \neq j}(λ_{k} P_{kj}P_{ik}(t)) - λ_{j} P_{ij} $$
Mattias R
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Poisson Process conditional probability

Let $N(t)$ be a Poisson Process with rate $\lambda$ Find $\displaystyle P(N(4) \le 2N(2) \mid N(2) = 1) = \frac{\sum_{i = 0 }^2P(N(4) = i, N(2) = 1)}{P(N(2) = 1)} = ?$ Can I split this up using the independent increments?
all.over
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