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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9 views

Is there something like a sampling theorem for Cox processes?

I'm considering an inhomogeneous Poisson process where the intensity function is modelled as a non-negative link function of a Gaussian process realization. This is a special case of a Cox process. ...
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how to simulate the Poisson point process on a irregular part of a sphere surface? [closed]

I've got a sphere and has a target area on the sphere, the target area is a polygon. Knowing the Poisson point density and the size of the area, I known the number of points which should be deployed ...
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How to compute $P\left(\max_{{M_s + 1} \leq j\leq M_u} Y_j < \max_{1\leq j \leq M_s} Y_j\right)$?

Assume that $M_t$ follows a Poisson process and $T_i,\quad i=1,2,\cdots$ and $Y_i, \quad i=1,2,\cdots$ are i.i.d. random variables showing the arrival of events and interarrival times of the process $...
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What is the difference between finding the hazard function by using an NHPP and fitting a Weibull curve to the data for a repairable system?

When modelling a repairable system, one should use a non-homogeneous Poisson Process (NHPP). E.g., the moment the hazard function I use in the NHPP is a Weibull function. However, I have difficulty ...
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81 views

Expected waiting time until no event in $t$ years for a poisson process with rate $\lambda$

As an example, assume that the yearly rate of earthquakes in a region is $2$, so on average two earthquakes happen every year. How long should I expect to wait until I expect to see no earthquakes in ...
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27 views

Waiting time for specific event in multiple independent poisson processes

If I have $n$ independent Poisson processes with rates $r_1, r_2, \ldots, r_n$, and $\sum_{i=1}^n r_i = r_{tot}$, the expected time until any event occurs is $\frac{1}{r_{tot}}$. If I am interested in ...
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Why does my simulation of nearest neighbors as circle origins provide different non-contact probability compared to theoretical?

I am simulating spatially distributed points in $\mathbb{R}^2$ with intensity $\lambda$ (units 1/area), which act as circle origins with radii being a random variable $R_k$. Given the distance to the $...
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1answer
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Distribution of number of events that occur a certain time between each other

I'm modelling an experiment in which events happen homogenously (i.e., Poisson process). The Poisson distribution models the distribution of the number of events that occur within $t$ of a particular ...
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1answer
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Predict background counts given past observations and assumption of linear variation of the rate parameter in time

Consider the following problem. We have a time series of counts (poisson-distributed) data. In this time series we can select an off-pulse window in which only background is present and a subsequent ...
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Testing if multiple independent low-rate counting processes are poisson

I have multiple independent counting processes and I want to check if they generally behave like Poisson processes. For the sake of the example, imagine a city with $N$ citizens and $M$ shops, where ...
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Increase weight of a certain outcome in a poisson distribution

I have no background in math, so excuse me if the answer to this is straightforward. I'm trying to build a model that predicts the outcome probabilities of a football game. According to the model, the ...
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Sequential Testing of Poisson Process

My question is related to the paper "Sequential Testing for Poisson Processes" by Peskir and Shiryaev, available here. Specifically, suppose that there are two states of the world, G(ood) ...
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Order statistics and Poisson process

Let $N(t)$ be $PP(λ)$.Given that $N(t)=n$, compute the probability of a) Last event before $t$ occurs before $3t/4$. b) First event after $t$ occurs after $t+h$, $0<h$. c) $...
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Parameter of exponential distribution

I am reading a paper about spam detection on Twitter and I want to check whether my interpretation is correct. We are given the Twitter accounts of users who post different URLs in their tweets. The ...
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The expected value of waiting time

A two-way street (i.e. vehicles pass from left and right). The number of cars coming from the left side follows a Poisson distribution with $\lambda$= 20 cars per minute, and the number of cars coming ...
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Cutoff for a poisson-gaussian mixture model

I have count data that is bounded on one side at zero (see image). It is bimodal and I think it results from two different processes. I would like to fit a poisson distribution around the hump around ...
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help with a question on Poisson process

Suppose events occur as a Poisson process with a mean rate of 4 per minute.I want help in the following: 1.Given that 5 events have occurred in 6 minute interval ,what is the probability that 3 events ...
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35 views

Poisson process - expected reward until time t

The calls to the fire department occur according to a Poisson process with a rate of three per day. The fire department must respond to each call. Of the calls that come into the fire department, on ...
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Distribution for the predictive number of events when arrivals come according to a Poisson process

Let's assume individual arrivals follow a homogeneous Poisson process of parameter $\lambda$. Assume also that these individuals can undergo an event following cumulative distribution $F$ thereafter. ...
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Non-homogeneous Poisson Process conditional problem

buses arrive at a bus stop according to a non-homogeneous Poisson process with the rate function $$\mu(t) = 1 + t$$ I have some idea on how to do this but I don't know if I'm correct. Q1: Assuming the ...
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Poisson process - time of arrival of a client, given that one client arrived at first interval

I have the following situation: I have a Poisson Process with $λ=7$ (seven customers / hour). This process describes the arrival of customers in a store. The store is open from 9:00 AM to 19:00 PM. My ...
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Time distribution of a Poisson Process

My problem is the following: I have a Poisson Process with $\lambda = 7$. This process describes the arrival of customers in a store. Now, I must find: The time distribution when the second customer ...
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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 ...
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Limiting distribution of $M/M/2/5$ queue with two heterogenous servers

A queue with a total capacity of $5$ customers has $2$ servers who serve at rates $\mu_1 = 1$ customer/hour and $\mu_2 = 2$ customers/hour respectively. Service times are exponentially distributed. ...
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Could you estimate the probability of arrivals of a poisson process?

Let $T_i$ ~ $exp(\lambda)$ be i.i.d exponential random variables, with unknown $\lambda$. These are the time intervals of the poisson process. And $X_n=\sum_{i=1}^nT_i$, are the arriving time of the ...
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259 views

Densities of Arrival Times of Poisson Process

People arrive at a store as a non-homogeneous Poisson process with rate t, where t is the time measured in hours between noon and 6pm. If we know that precisely one person arrived in the first hour, ...
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1answer
137 views

Probability of compound Poisson process

Let $X$ be a compound Poisson process with rate $\lambda$ and increments $Y_i = \pm 1$ with probability $\frac{1}{2}$. Find $P(X(t) = 0)$. I tried conditioning on $N(t)$: $$ P(X(t) = 0) = P(\sum\...
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1answer
36 views

Renewal process vs inhomogeneous Poisson process?

I am analyzing a dataset with recurrent events, and considering two candidate models: A model based on a renewal process (time is measured from the previous event) A model based on an inhomogeneous ...
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3answers
80 views

Probability of drawing two differently colored balls in a partition of equally sized partitions

If I randomly distribute, say, 100 red and 10 blue balls onto a field of 10000 square meters (100*100 m), how would one calculate the chance of having at least one red and at least on blue in the same ...
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Estimate partially observed Poisson process

I try to estimate the intensity of a Poisson process $P_1$, but it is not fully observable. There are some "obervers" coming to the system which follow another Poisson process $P_2$. In ...
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1answer
36 views

How to Choose Poisson Time Interval

A Poisson process is one where mean = var = λ. How do you decide what time interval fulfills these criteria when fitting the Poisson distribution to a process? Can all processes be modeled as Poisson ...
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Question about solution: Poisson process & conditional expectation

Given the following problem: Alice shows up at an Athena cluster at time $0$ and spends her time exclusively in typing emails. The times that her emails are sent are a Poisson process with rate $\...
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Key elements of a Poisson Process? [duplicate]

I'm new to Stochastic Processes, in general. However, I see that they come in varying forms of complexity. A gaussian process (and gaussian process regression) are quite complicated and can easily fit ...
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102 views

MLE for Poisson Process

I asked a similar question here regarding method of moments, and now I need to figure out how to solve using MLE. Here is a repeat of the problem: A School of Ornithology researcher wants to estimate ...
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1answer
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Poisson Process Method of Moments

Disclaimer: This is a homework problem A School of Ornithology researcher wants to estimate the number of red-tailed hawks in Ithaca. She radio tags 10 birds, and then sets up a feeding station with ...
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How to make sense of rescaling time series of counts?

I'd like to forecast time series of counts : sold items. Each time series represents monthly sales. I also believe that there are clusters within the series, with low, medium and high count items. ...
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1answer
303 views

Sample size for binomial distribution for rare events

I would like to estimate the parameter p in a binomial process but need to determine the sample size required. I believe the event is fairly rare, and I have read the normal approximation may not be ...
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37 views

Simulating non-homogenous Poisson Process with piecewise constant intensity rate

Suppose we have a non-homogenous Poisson Process with the following intensity rate: $$ \lambda(t)=\left\{\begin{array}{l}5, t \in(1,2],(3,4], \cdots \\ 3, t \in(0,1],(2,3], \cdots\end{array}\right. $$ ...
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What is the distribution of the k-th event in the Poisson process? [duplicate]

Assuming we have a Poisson process of density $\lambda$ I'm trying to find a distribution of a random variable $\tau_k$ - the time when k-th event has happened. E.g. in case of $\tau_1$ it is an ...
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289 views

Can Negative Binomial parameters be treated like Poisson?

I have a count process that I'd like to model with a Poisson process. Data is measured every 30 minutes, and with a poisson distribution I can easily measure the probability of a given count of events ...
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1answer
64 views

Poisson distribution marginal probability of sufficient statistic

I am self studying a theoretical statistics course I found online. There is a question to show that for $(X_1, ... X_n)$ i.i.d Poisson variables with parameter $\theta$, the statistic $T=\sum_{i=1}^N ...
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63 views

Simulate Non Homogeneous Poisson Process

I've some problems with simulating arrival times following a non-homogeneous Poisson process. I am using the following arrival rate function: $\begin{equation*} \label{eq:lambda} \lambda(t) = \left\{...
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Standard Deviation of Bernoulli trials / Bernoulli process with different probabilities

If 20 independent Bernoulli trials are carried out each with a different probability of success and therefore failure, what is the standard deviation of the result? How is this calculated?
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$M/M^B/1$ Burke's theorem : what is the distribution of the output batch interarrival times?

Setup: Take an M/M/1 queue: the inputs arrive according to a Poisson process at rate $\lambda$, the service time per item is distributed exponentially with mean $1/\mu$, $\mu > \lambda$ the ...
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1answer
61 views

Poisson Process Conditional Probability computation

Given $N(t)$ a Poisson Process with generation rate $\lambda$ with $t_1<t_2$ and $N_2>N_1$ I'm looking for a way to express the following probability: $$ P[N(t_2)>N_2|N(t_1)<N_1]$$ In ...
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Likelihood of a truncated (in)homogenous Poisson Process

I am studying a Poisson process of events that have occured until a given time. Theses occurences $T_n $a re observed by the observer only if the delay $U_n$ between occurence and time of report ...
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Distribution of the ratio of two poisson processes

I have two Poisson counting processes that generate events of type 1 or 2 at some independent rates $r_1$ and $r_2$. In a fixed time interval $T$ I count $N_1\sim\textrm{Poisson}(r_1 T)$ type 1 events,...
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1answer
59 views

What length are some segments of a broken rod?

If a rod (of unit length) is broken into $n$ segments (assuming the $n-1$ breaks occur with uniform probability across the entire length) and $k$ of those segments are chosen at random and laid end to ...
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102 views

How to estimate the intensity rate $\lambda$ of a Cox Process

In a Cox Process, or doubly stochastic Poisson process, the intensity rate itself is a stochastic process that varies across space or time. Let us assume that the intensity rate has the following form ...
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1answer
62 views

How to correct for unequal observation time in poisson regression

So I have data from a number of people on the number of sexual partners they've had in their life so far (self reported, but let's assume they're perfectly accurate for this question). I wanted to see ...

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