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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Poisson Distribution Process Control Limits

I am measuring the number of events that happen within n minute intervals for a group of 100 stores. My problem is, I am looking for a way to calculate a threshold for the acceptable number of times ...
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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 ...
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99 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 ...
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384 views

Is it valid to simulate a shot noise Poisson process with discretization (R)

I have a two-factor shot noise process as follows (sorry for the picture of text): I want to simulate from this process. Is it valid to do so in a discretized way? The solution to these SDEs is a ...
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36 views

Homogeneous poisson process and identical distribution

Suppose that the arrival of men and women at a bank can be considered independent homogeneous Poisson processes with mean 10 and 8 at every 30 minutes, respectively. In a interval of 10 minutes, what ...
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49 views

Why isn't there a bound on the waiting time for the first occurrence in Poisson distribution?

From my book: Let $W$ denote the waiting time until the first occurrence during the observation of a Poisson process in which the mean number of occurrences in the unit interval is $\lambda$. ...
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200 views

Simulate Homogeneous Poisson where each event is uniform

The problem is the following: Buses arrives according to a Homogeneous Poisson with arrival tax of 5 per hour. Each bus can contain 20, 21, 22... 40 passengers with eaqual probability. ...
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5k views

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 /...
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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 ...
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1answer
247 views

Time Rescaling Theorem and Residual Analysis

Let $\mathcal{P}$ an homogeneous unit rate Poisson process. It's conditional intensity function (star indicating conditioning on the history) can be written as $$\lambda^*(t) = \lambda = 1$$ meaning ...
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161 views

Howto derive statistical upper limit in case of zero observation of poisson process?

I am writing software, that keeps track of hourly rates of certain incidents in historic data. I find that the generating process of these incidents is acceptably well described by a poisson process. ...
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324 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 $ ...
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3k views

Likelihood Ratio Test for Poisson Distribution

Suppose we have the following count data: ...
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52 views

Calculating the p-value of two independent counts? [duplicate]

My simple experiment: Two equally-sized patches of the night sky are observed: Patch A contains $100$ stars Patch B contains $110$ stars My null hypothesis is that stars are randomly distributed in ...
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232 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 ...
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245 views

Probability of a random occurrence of Poisson process to be the first occurrence

It is given, that there is an occurrence of Poisson process (at time $t$) of intensity $\lambda$ within an interval $(0, T)$. $t \in \{t_i\}; 0 < t_1 < t_2 < \cdots < t_n < T$ I need ...
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26 views

Total hourly profit for a single-server food stand

Customers arrive to a single-server food stand according to a Poisson process with rate $20$ per hour. The time to serve a customer is exponentially distributed with a mean of $2$ minutes. (a) The ...
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177 views

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 ...
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301 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 ...
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134 views

Evaluation metrics for cluster or cox process

I am working with spatial point processes and on a dataset which seems to be a non-homogeneous poisson point process. I have fitted a cluster/ cox process model and also used this model to predict the ...
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1answer
2k views

Distribution of inter arrival times in a Poisson process

I am new to Statistics. I am studying Poisson process, I have certain questions to ask. A process of arrival times in continuous time is called a Poisson process of rate $\lambda$ if the following ...
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37 views

Expected ratio of right-censored to non-censored waiting times in Poisson proces

I am trying to calculate the expected value of $Y = \frac{\sum_{i=1}^n min(x, X_i)}{\sum_{i=1}^n X_i}$, where $X_i$ are independent random variables of exponential distribution of ratio $\lambda$. I ...
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109 views

Poisson and Negative Binomial Models - Are real counts really necessary?

The Poisson model is probably the first that pops into mind when trying to model count data, which is generally described as: non-negative integer data. The Poisson distribution is defined as modeling ...
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134 views

Test if one Poisson process's rate is smaller than a given value

Suppose $N(t)$ is a Poisson process with rates $\lambda$. Suppose I've been observing it for $t \in [0, T]$ and recorded events. How can I test the null hypothesis $\lambda < \lambda_0$, where $\...
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168 views

Interarrival times of Negative Binomial (Yule-Furry) process?

The interarrival (ia) times of a Poisson process are exponentially distributed. What distribution function describes the ia times of a Negative Binomial process aka Yule-Furry process? Thanks!
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1answer
188 views

NIST exponential distribution Poisson distribution

An exponential distribution describes the time between events in a Poisson process. Suppose that the average waiting time for an action is 5 minutes. The time waited each time measured in hours is an ...
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115 views

Alternatives to Poisson Distribution

Imagine a data set that has information about how frequently various people visit a location $l$ over a year ($l$ could be a restaurant or a public park). What would be a good way to analyze this data?...
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135 views

Integral of a continuous family of i.i.d bernoulli random variables

Let $F(t) \sim \operatorname{Ber}(p)$ for every $t \in [0,1]$. Let $X = \int_0^1 F(t)\,dt$. $X$ is of course itself a random variable. Questions: Does $X$ exist? If so, what is the distribution of $...
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Memoryless Property

Here is question I am working on to study for an exam that I am quite not sure how to frame a proper answer for: Trains arrive at a station with i.i.d interarrival times following an exponential ...
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2answers
108 views

Poisson probability of more than 200 events

The $N(t)$ is a poisson process for the number of events to occur with a mean $\lambda(t) = 3$ per day. I am supposed to find the probability of more than 200 events in 60 days. My theoretical answer ...
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459 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%; ...
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1answer
741 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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How do I evaluate the accumulated error of a repeated Poisson process?

I'm quite newbie in stats, so sorry if I write anything stupid. I'm dealing with a statistical process in my master in electronics and I'm kind of stuck. Consider a device that counts individual ...
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108 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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121 views

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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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. ...
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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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1answer
986 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 ...
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225 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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782 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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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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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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640 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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1answer
208 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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Modeling Arrivals With a Time Limit

I have some data (sample here): ...
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1answer
166 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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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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88 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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52 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/...