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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Please explain the waiting paradox

A few years ago I designed a radiation detector that works by measuring the interval between events rather than counting them. My assumption was, that when measuring non-contiguous samples, on ...
45
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14k views

Is there any gold standard for modeling irregularly spaced time series?

In field of economics (I think) we have ARIMA and GARCH for regularly spaced time series and Poisson, Hawkes for modeling point processes, so how about attempts for modeling irregularly (unevenly) ...
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How to know if a data follows a Poisson Distribution in R?

I am an undergrad student and I have a project for my probability class. Basically, I have a dataset about the hurricanes that impacted my country for a series of years. In my probability Book, (...
24
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1k views

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
8k views

How to estimate Poisson process using R? (Or: how to use NHPoisson package?)

I have a database of events (i.e. a variable of dates) and associated covariates. The events are generated by the non-stationary Poisson process with parameter being an unknown (but possibly linear) ...
12
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1answer
11k views

What are the differences between survival analysis and Poisson regression?

I'm working on a classical churn prediction problem using the number of visits of a given user to a site and I thought that Poisson Regression was the right tool for modelling the future engagement of ...
10
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1answer
10k views

How do we predict rare events?

I am working on developing an insurance risk predictive model. These models are of "rare events" like airline no-show prediction, hardware fault detection, etc. As I prepared my data set, I tried to ...
9
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1answer
228 views

Probability of an independent Poisson process overtaking another

I have asked this question before in another fashion on other stackexchanges, so sorry for the somewhat repost. I have asked my professor and a couple of PhD students about, without a definitive ...
8
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2answers
566 views

Total expectation theorem for Poisson processes

I have two independent Poisson processes $A$ and $B$ with arrival rates $\lambda_A$ and $\lambda_B$, respectively. Now, the expected time for the arrival of the next item for the merged process should ...
8
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2answers
1k views

Are there any alternatives to simulation for determining the distribution of number of events from two dependent non-homogeneous Poisson processes?

A "state of the art" model for the distribution of goals scored in a soccer match is that of Dixon and Robinson (1998) "A Birth Process Model for Association Football Matches" which accounts for two ...
7
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2answers
1k views

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 ...
7
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3answers
300 views

Paradox of Poisson process with at least one event in the interval

Let $X_T$ is a number of events in Poisson process of unitary rate ($\lambda = 1$) within interval of length $T$. It is known that at least one event has been observed in the interval, I want to find ...
6
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1answer
258 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 $N$ ...
6
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2answers
209 views

Motivation for gamma distribution with a non-integer parameter

The Erlang distribution has a straightforward interpretation in terms of waiting time for the occurrence of a predefined number of events in a Poisson process or a sum of a predefined number of ...
6
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1answer
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Poisson process and the memoryless property

I understand that inter-arrival times of a Poisson process are exponentially distributed and therefore the inter-arrival times are memoryless. However, how about the waiting times of Poisson process ...
6
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1answer
268 views

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 ...
5
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1answer
378 views

Regression for poisson process in R

I have a series of samples of varying length, and the number of bugs created in those time samples. Reading the literature, this is often modeled as a Poisson process. If you write it like: $$P(k,t)=\...
5
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2answers
1k views

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. ...
5
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1answer
148 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 $ \{ ...
5
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2answers
330 views

Number of events of type B given n events of type A in a Poisson process

I have a simple probability calculation I was working on that I came up with an answer for, but a question a colleague asked me led me to come up with a second approach - and a different answer. I put ...
5
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3answers
315 views

examples on sequence of poisson random variables

Let $X_n$ be distributed as a poisson random variable with parameter $n$. Then which of the following are true ? 1.$\underset{n\rightarrow \infty}{\lim} \mathbb{P} (X_n > n + \sqrt n)=0 $ ...
5
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1answer
494 views

Estimating Poisson process intensity using GLM

Suppose I want to build an explanatory model for events generated by an inhomogeneous Poisson process with unknown intensity $\lambda$. Each entry in my dataset represents the registration of an event:...
5
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2answers
456 views

Poisson distribution problem - traffic problem

Hi So I have this question below. I know I need to model each lane as a separate Poisson distribution. The possible answers are: a) 11.4%; 22.4%; 33.4%; 44.4%; 55.4% b) 2.74%; 4.74%; 12.74%; 34.74%; ...
5
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1answer
646 views

The gamma distribution and Poisson processes

I know that the gamma distribution with parameters $k$ and $\theta$ can be used as a model for the occurrence of events. The requirement on the events would be that their occurrence is random and the ...
4
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1answer
736 views

Compound Poisson random variable

A compound Poisson random variable $S$ is defined as: $S=\displaystyle\sum^N_{i=1}X_i,$ where $N$ is a random draw from a Poisson distribution with intensity parameter $\lambda$, and $X_i$ are ...
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2answers
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Help me understand poisson.test?

I want to understand the poisson.test() function: ...
4
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1answer
11k views

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 ...
4
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1answer
401 views

Poisson processes

I have two realizations of a poisson stochastic process, they are over the same space with rate $\lambda_{1}$ and $\lambda_{2}$. What is the probability that N elements in both sequences are the same, ...
4
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1answer
1k views

Distribution of arrival times to server for an M/M/1 queue (what the server experiences)

In an M/M/1 queue, we know that inter-arrival times are exponentially distributed, and that service times are the same. What is the distribution of to-server inter-arrival times (aka service start ...
4
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1answer
97 views

Testing Poisson process where $X(t)$ is given at fixed times

I have a discrete stochastic process $X(t)$ which I believe is a Poisson process, that is the value of $X(t)$ at time $t$ is a Poisson random variable with parameter $\lambda t$ and disjoint intervals ...
4
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1answer
963 views

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 ...
4
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1answer
416 views

Mean service time of a $M/E_2/1$ queueing system?

Consider a $M/E_2/1$ queueing system, where the customer arrival rate is $\lambda$ and the service time distribution has a gamma distribution with parameters $2$ and $\mu$, i.e. with p.d.f. $\mu^2te^{-...
4
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1answer
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Testing for Poisson process

I have some discrete times of events and I would like to do a test to see if they are likely to have come from a homogeneous Poisson process. From this pdf, I see: REMARK 6.3 ( TESTING POISSON )...
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2answers
2k views

Likelihood of multiple event times modeled as independent Poisson processes

I am modeling three events A, B, and C as Poisson processes with rates $\lambda_A$, $\lambda_B$, and $\lambda_C$ and I would like to calculate the likelihood of observing some data given my model. A ...
4
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1answer
367 views

Expectation and confidence intervals of a Poisson process

A Poisson process has PDF $$P(X=k)=\frac{e^{-\lambda t}(\lambda t)^k}{k!}$$ I'm trying to find an expression for: $E[X | \lambda, t]$ Confidence intervals (i.e. find $\delta$ such that $P(\bar{x}-\...
4
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1answer
88 views

Mean length of time spent queueing in $M/E_2/1$ system?

Context: Consider a $M/E_2/1$ queueing system, where the customer arrival rate is $\lambda$ and the service time distribution has a gamma distribution with parameters $2$ and $\mu$, i.e. with p.d.f. $\...
4
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1answer
443 views

Test hypothesis point process is Poisson [duplicate]

I have some data and I would like to test the hypothesis that they come from a homogeneous Poisson process. I can of course look at the inter event times and test if these are exponentially ...
4
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0answers
114 views

Detecting changepoint in ratio of rates of two Poisson processes

I'm interested in a changepoint detection problem of the following scenario: Consider two Poisson processes for which we have the event times. I'm interested in detecting a change in the relative ...
4
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0answers
234 views

Probability of collisions in queues of Poisson process

I have a process whereby objects of width $W$ land on a gene at rate $F$ (per second, poisson process, lets assume), and then start to move along at constant speed $V$. I'm trying to work out the ...
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5answers
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How to simulate Poisson arrival times if the rate varies with time?

Suppose we are to study a non-homogeneous Poisson process of 3 hour cycles in which: At the first hour, the arrival rate is 1.5 events / hr. At the second hour, the arrival rate is 2.1 events /...
3
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2answers
54 views

Alternating between two states {A, B} each with exp distributed durations. What's the probability of state=A at time t?

Say I have a light bulb that can be on (A) or off (B). It alternates between being state A or B. It will be in state A for a duration a ~ exp(α), and in state B for duration b ~ exp(β), (...
3
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2answers
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Simulation of a Poisson Process

I am trying to simulate the compound Poisson process using the next algorithm that I found in a textbook on stochastic processes. Let $S_0 = 0$. Generate i.i.d. exponential random ...
3
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2answers
910 views

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 ...
3
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2answers
66 views

How to quantify the reliablility of an estimate

Lets say I inspect two workers. Worker A works one hour and produces 2 goods. Worker B works 100 hours and produces 180 goods. So, on average, worker A produces 2 goods per hour and worker B 1.8 ...
3
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2answers
87 views

Any expression for the probability of a hard sphere in Boolean model

I am working hard on a problem of Boolean model. In a example of Boolean model, points are scattered in the plane according to a homogeneous Poisson process of intensity λ. On each of these points a ...
3
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1answer
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Comparing different poisson distributions with very variable sample sizes

I have data on around 50 different roads: The number of accidents and the volume of traffic on each. I'd like to compare these and estimate which is the most dangerous/safest, but the volumes (and ...
3
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1answer
2k views

Maximum likelihood for number of events in Poisson process

I have a Poisson process with parameter $\lambda$ known. How do I compute the maximum-likelihood estimator for $N$, ie. the number of events over a specific time spell $T$. To repeat, I know $\lambda$...
3
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1answer
1k views

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 ...
3
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1answer
522 views

Poisson Process

I would appreciate a hint on this problem: A pedestrian wishes to cross a single lane of fast-moving traffic. Suppose the number of vehicles that have passed by time $t$ is a Poisson process of rate $...
3
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1answer
1k views

Prior for Bayesian Inference on Failure Rate in Poisson Distribution

I'm trying to derive the posterior distribution for the failure rate (lambda) of a process with poisson distribution. I have tried the use of an improper uniform distribution on lambda by letting the ...