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.

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116 views

Burke's Theorem for rejection from Erlang-B loss queue

I have some general uncertainties regarding the rejection process from an Erlang-B loss queue ($M/M/c/c$), where the total capacity of the queue is equal to the number $c$ of servers. Consider the ...
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Marking a hard core poisson process

A hard core process (HCP) deals with the deposition of hard spheres, generally of the same radius, that are forbidden to overlap. Suppose instead that the identical spheres are replaced with distinct ...
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Proof for the distribution of the increments of Gaussian and Poisson processes?

What is analytic proof for the fact that increments of a Gaussian process are again Gaussian? How could I apply that to increments of Poisson process? Maybe do I multiply characteristic function ...
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Characteristic Function of a Compound Poisson Process

The definition of a compound Poisson process and its characteristic function I have are the following: Let $\lambda>0$ and $N\sim\text{Poisson}(\lambda T)$. Also, $\{X_i\}_{i=1}^N$ are i.i.d. ...
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How do I check whether a series of time events does not show acclimatization? And test it for several animals. R

I am studying animal behavior and I want to know whether the frequency of a specific behavior changes with time. I thought it was a common and simple problem but I cannot find papers with examples ...
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Poisson Distribution: Estimating rate parameter and the interval length

Here is the motivation for my question. I have a sensor that reports data to me. The occurrence of the reports from the sensor follows a Poisson process (so, obviously, the inter-event times are ...
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246 views

Conditional distribution (on N) of arrival times in a nonhomogenous poisson process

Conditional on $N(t)$, given some $\lambda(t)$ characterizing some Nonhomogenous poisson point process, the distribution of an arrival time $t_i$ is $\lambda(t_i)/\int_{A}\lambda\left(t\right)dt$ ...
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818 views

Probability Distribution for Inter-arrival Time

I'm trying to build a simulation for a quality control process, where quality analysts inspect the product and report faults if they found any. I have a dataset of this bug reports, so I'm trying to ...
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93 views

About Garman's inventory model

In Garman's inventory model (http://www.sciencedirect.com/science/article/pii/0304405X76900064), buying order and selling order are Poisson processes with order size = 1. Buying price and selling ...
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1k views

Age and residual life time of the Poisson process

Original Question Let $N(t)$ be a Poisson process with intensity $\lambda$. Let $T_1<T_2<...$ be the occurrence times. Let $T_0=0$. For any $t>0$, define the $age$ random variable to be $...
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665 views

Predicting intensity of Poisson process, given event data

I have a dataset of events: each row is an event, and each column is a feature. There are millions of events and several dozen features. The features are mostly numerical (a few are categorical and I ...
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226 views

Question about marked poisson process

Let's say I have a Poisson point process on $\left[0,T\right]$ with rate $\lambda\left(t\right)=2t^2$. Suppose I attach a mark $m_t$ to each point $t$ of the process such that $m_t\sim N\left(t,1\...
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49 views

Modeling Arrivals With a Time Limit

I have some data (sample here): ...
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173 views

Problem on Poisson Process

I am doing some problems related with the Poisson Process and i have a doubt on one of them. The problem is stated as follows: A doctor works in an emergency room. The emergencies arrive according a ...
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151 views

Find the distribution of $ N = \min \left\{k: \prod_{i = 1}^{k}U_i \lt .6\right\}. $

I'm cross-posting this from math.SE because it's not getting any love over there. However, if that's considered heresy, I can delete the posting over there. The Statement of the Problem: Let $ \{ ...
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92 views

Maximum value in Poisson process investigated using scan statistics

We have process where events are occurring at a rate of $B$, where the distribution of events in a fixed time frame can be described using Poisson statistics. Thus, the events can be modeled using a ...
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53 views

Decomposing multiple poisson process

Assume a time series composed by many recurring events coming from many different poisson process each with a different rate. Lets assume for simplicity no overlap between events. Is there any math/...
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289 views

Time up to $n$th event in Poisson process distributed as $\frac{1}{2\lambda} \chi^2_{2n}$

Let's assume that a number X of some events over time $t$ is modeled by Poisson distribution with rate $\lambda$ (here, it's rate, not mean): $$ X \sim Poisson(\lambda \cdot t) ~~~~ (\lambda t ~\text{...
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383 views

Poisson process for queuing problem

I posted it incorrectly on a different thread so I'm reposting it here. I'm working on a problem and would love to get any advice - patients come into a clinic according to a Poisson process with ...
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Switch from Modelling a Process using a Poisson Distribution to use a Negative Binomial Distribution?

$\newcommand{\P}{\mathbb{P}}$We have a random process that may-or-may-not occur multiple times in a set period of time $T$. We have a data feed from a pre-existing model of this process, that provides ...
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1answer
381 views

Probability of a rare event

Let's say I consider an event rare if it occurs no more than once in 90 days. Assuming everything is random and independent, If I see this event on day 3 of the observation, what is the probability ...
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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 ...
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291 views

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 $N$ ...
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173 views

Poisson distribution of rain storm arrival

I know what poisson distribution means.But I can not just understand how rain cells or rain storm arrivals is poisson process? looking for simple explanation. Thanks in advance
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Odds of specific generated population of exponential distributed stochast

I'm trying to generate a sequence of samples using an exponentially distributed stochast, i.e., making a Poisson arrival process. In my specific case I generate 337 samples using a mean inter-arrival ...
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1answer
99 views

Confidence interval for non-homogeneous Poisson process where lambda is fourier

I have a non-homogeneous Poison distribution $X = \{ x_1, x_2\, .., x_n\}$ where: $$\lambda(x) = \exp(a_0 + \sum_{z=1}^{Z}(b_z \sin(2 \pi x \frac{z}{Z}) + c_z \cos(2 \pi x \frac{z}{Z})))$$ $$\Lambda=\...
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183 views

Expected Value in Poisson Point Process with Prior Knowledge

I have a setup with a homogeneous Poisson Point Process (PPP) of intensity $\lambda$ in $W \subseteq \mathbb{R}^d$ and a set $A \subseteq W$. I'm looking for the expected value of points in set $A$, ...
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120 views

How do you calculate the mean and variance of a random var with a distribution function that has a parameter with its own distribution function?

I am busy with ruin theory. $$ S(t) = \sum_{i=1}^{N(t)} X_i $$ $S(t)$ is the aggregate claim size after $t$ years, where $X_i$ is the individual claim size (with mean and variance given) and $N(t)$...
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513 views

distribution of the last arrival in poisson process

Consider a Poisson process with rate $\lambda$ and let $L$ be the time of the last arrival in the interval $[0,t]$, with $L=0$ if there was no arrival. How can I prove that t-L has exponential ...
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Expected number of times two Poisson process events occur on the same day

So I have set up a Poisson Process N(t) with parameter L (events/year). I want to find the expected number of times over a 3 year period that 2 events occur on the same day. My approach: First ...
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43 views

Wait time for non-homogenous Poisson processes

Is the wait time between events for a non-homogenous Poisson process still exponentially distributed
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321 views

Autocorrelated Inter-arrival Times of Extreme Events

I'm using a bunch of techniques and methods from Extreme Value Theory to analyze my data. I have a time series representing the number of events happening in a given day. The time series is unequally ...
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Estimate lambda for panel count data

I have panel count data for $F$ firms across $I$ years, so observe counts $C_{f,i}$ for $f \in \{1,...,F\}$, $i \in \{1,...,I\}$. I want to model the data as a poisson process. With increasing $C_{f,...
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192 views

Pmf of compound Poisson process

Can I obtain an analytic expression for pmf of compound Poisson process? $Y_t = \sum \limits_{i=1}^{Xt} D_i$ where $X_t$ ~ Poisson($\lambda$) and $D$ ~ Geometric($\rho$)
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Poisson vs. Gaussian in Geomagnetic Data

I've been studying geomagnetic signals using a threshold approach to detect pulse events in the data. The question here is what is the significance of the crossover of stddev and mean as the ...
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Alternatives to a one-dimensional Poisson process [closed]

Say I have "arrival" times in what may or may not be a Poisson process. I can think of at least three ways in which it can deviate from a Poisson process: Clumping. One arrival is likely to be near ...
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Prove that $N(\tau),V_t,Z_2,\ldots$ are independent in Poisson process

We define a Poisson process is a renewal process in which the interarrival intervals $X_n$'s have an exponential distribution with parameter $\lambda$. Denote $N(t)$ is the number of arrivals in $(0,t]...
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413 views

estimate confidence interval for poisson process

I would like to know how I can estimate the confidence intervals for poisson process distributed variables. I have a pandas dataframe with a column of trials and a column of successes. I want to ...
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761 views

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 ...
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Poisson Process in R from exponential distribution

I am trying to simulate a poisson process sample path in R by starting off with exponentially distributed random variables. For example, for a value of $\lambda=0.5$, I can generate 500 samples and ...
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Using Poisson process model for prediction

Suppose some event $X$ occurs on average $10$ times per minute. The events are independent of each other. Now, if I have understood correctly, this can be modeled as a Poisson process, and I can ask ...
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Overlapping interval of a Poisson arrival process

Calls arrives according to a Poisson arrival process with rate lambda = 15. Find E(N(2,4]N(3,5]) My thoughts: E(N(2,4]) = E(N(3,5]) = lambda * t = 15 * 2 = 30 However, I cannot figure out the next ...
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Is there a family of processes centred on the Poisson process?

I am looking for a model, characterized continuously by a single parameter, to describe the arrival times of buses with unit expected interarrival time. At one extreme of the parameter (say $\theta=1$)...
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A math proof within a question about homogeneous Poisson process

We know that a homogeneous Poisson process is a process with a constant intensity $\lambda$. That is, for any time interval $[t, t+\Delta t]$, $P\left \{ k \;\text{events in}\; [t, t+\Delta t] \right \...
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Help me out with real-life Gamma distributions

I'm working with some data relating to maintenance and parts failures. I've got a pure math background with only a little bit of probability, and I'm currently learning by doing. I've got a list of ...
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Multiple poisson processes ?

I'm just trying to get my head around Poisson processes, as they're fairly new to me, and I've had a thought experiment that has been annoying me a little. Imagine a volume of some mixture hit by X-...
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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 ...
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Expected value of a product of two compound Poisson processes

I'm working on my master thesis now and I've been struggling with a problem for some while now and no one seems to be able to help me or point me in any direction. So now I reach out to see if someone ...
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Uses of Poisson process in stock price models

If I want to find the probability that a stock is going to touch a support or resistance at least once in the next 5days, can I use a Poisson distribution? The textbook examples usually say that ...
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Split Poisson Process AND severity

I have a Poisson process whose statistics are interarrival times ($\bf X$), number of arrivals ($\bf N$), and arrival times ($\bf T$). Later, the process is split by a Bernoulli process that ...