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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.

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Question regarding Poisson process [closed]

The following question came in my previous year exam. Any help will be appreciated. Q. Let customers arrive at a departmental store according to a Poisson process with rate 10. Further, suppose that ...
TuHere's user avatar
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What do we call a Poisson point process with an instantaneous log-rate being a Wiener process?

I have implemented a stochastic process for simulating demand of service that wanders in its average rate. This is a useful scenario for evaluating a controller that tries to optimize availability and ...
Galen's user avatar
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13 votes
3 answers
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Is a Poisson minus a constant still a Poisson?

I am working with a process in which I expect my variable to be Poisson distributed. For reasons that have to do with the scale, however, the values I obtain have a minimum of 11. I have noted than ...
elcortegano's user avatar
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20 views

Exponential function for Poisson intensity

I'm getting confused on estimating the intensity for a Poisson process. My background in the subject is weak. Suppose I'm interested in modelling the probability of an event occurring given some input ...
rudinable's user avatar
2 votes
0 answers
38 views

Can any thinning of a Poisson point process be thinned further into a Poisson point process?

Is the following true? Let $\Phi$ be a homogenous Poisson point process of positive intensity. Let $X$ be a thinning of $\Phi$ where $X$ is stationary and has positive intensity. Then $X$ can be ...
PtH's user avatar
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3 votes
0 answers
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Can any point process be thinned into a Poisson point process?

Is the following true? For any point process $X$ there exists a procedure $\mathcal{P}$ and a (homogenous) Poisson point process $Y$ with intensity $\lambda > 0$ such that the process of drawing a ...
PtH's user avatar
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1 vote
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What is the distribution of needed hospital beds?

Suppose I am modelling a hospital service with $k$ number of beds. Initially there are $m$ number of beds being used, where $m \leq k$, each of which has a known amount of time that it has been ...
Galen's user avatar
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4 votes
1 answer
51 views

Intuitive explanation of paradoxical interval times distribution

When I simulate a Poisson process on the interval [0,1], then the interval time between successive points follows an exponential distribution. E.g. in the code below when I select ...
Sextus Empiricus's user avatar
1 vote
0 answers
38 views

How could I fit a model of a non-homogeneous Poisson process in STAN? [closed]

I have some data $t_1, t_2, ..., t_n$ where $0 < t_i < T$ for all $t_i$. I assume that this has been generated by an inhomogeneous Poisson process with parameter $\lambda(t)$ defined again for $...
user1747134's user avatar
2 votes
0 answers
25 views

Modelling consecutive Poisson point processes, where the second starts only after the first has x counts

I'd like to model a situation where there are 2 consecutive Poisson point processes with different event rates, and the second process only starts after the first process reaches a cumulative event ...
abcoxyzide's user avatar
1 vote
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Distinguish whether time series comes from or another Poisson process

I have a time series of binary events, where the events are drawn from a Poisson process with a 50% chance of having a rate of $\lambda_1$ and a 50% change of having a rate of $\lambda_2$. Here $\...
KHAAAAAAAAN's user avatar
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Is there a correction for samples from a (linear) Prophet model when trained on an inhomogenous Poisson point process?

Facebook's Prophet is a popular modelling choice for time series forecasting in production due to many steps being automated (and thus convenient). This can sometimes lead to over-reliance on it when ...
Galen's user avatar
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1 vote
0 answers
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The Probability of an Earthquake Event

I'm looking for ways to to find the probability of an earthquake event from Twitter posts. I came across an equation in a research article that I need to understand and use. My goal is to write a ...
ORDeSUrv's user avatar
2 votes
1 answer
68 views

Reversible-jump MCMC and Poisson processes

Suppose we have a time interval $t \in [0, T]$ in which events occur as a Poisson process with some arbitrary time-dependent rate $\lambda(t)$. These events occur at times $Y=(Y_1, Y_2, \dotso, Y_M)$ ...
Jordan's user avatar
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2 votes
0 answers
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Total N patients for precision around a rate, with varying FU times

To design a retrospective longitudinal study (outcome: certain event after drug exposure), we know that the annualized rate of that event is 0.06%, aka 0.0006/person-year (possibly underestimated as ...
Alex Ortega's user avatar
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Poisson distribution in two disconnected region

Suppose $N$ is a RV such that$$N\sim Pois(\lambda)$$ and $N_{1}$ and $N_{2}$ are two Random Variable distributed in two disconnected region called $R_{1}$ and $R_{2}$ respectively and $N=N_{1}+N_{2}$ ...
Rust32627's user avatar
2 votes
0 answers
27 views

What is the appropriate normalization for finding correlations between Poisson distributions? [closed]

I am interested in using this algorithm, glm-pca, to find a lower dimensional embedding in time series data, specifically neuronal spiking data, which is Poisson distributed. I have looked at some ...
Angus Campbell's user avatar
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Expected arrival time for the first person in the decomposition of poisson process

Question: Customers arrive at a shop according to a Poisson process with rate $\lambda > 0$ per hour, with probability $p$ being male and $1-p$ being female. During the first hour $n$ people ...
Frank Lee's user avatar
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0 answers
26 views

Calibrating a non-homogeneous Poisson process to my data [duplicate]

My question: Let's say I have some data on the cumulative number of infections per day since the start of a pandemic at $t=0$. Since clearly the infection rate changes over time, I want to calibrate a ...
Michaël's user avatar
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What is the optimal measuring time split for limited measuring time between signal+background and background in a Poisson counting experiment?

I’m trying to figure out the best split of time between measuring either background or signal+background in a counting experiment in the case where we have prior estimates for the signal and ...
Physicist_285's user avatar
1 vote
1 answer
56 views

Statistical test whether data conforms to a spatial point process--gaza bombing locations

I came across this image on twitter, and it made me think about testing a point process hypothesis. Now this is a politically sensitive image, and I don't want to run afoul of any SE posting ...
krishnab's user avatar
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7 votes
1 answer
185 views

Are these two equivalent forms for the likelihood of a Poisson point process?

I have a Poisson point process in a bounded region $W$. I'm trying to calculate the likelihood of observing a particular set of points within $W$. I'm told that there are two equivalent forms of ...
The Pointer's user avatar
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Distribution of a process

I'm having trouble on solving this problem. Given a Poisson process X with parameter lambda and, indipendently, a random variable T with density f(t)=$\theta*e^{(-\theta t)}$, compute the ...
Onofrio Olivieri's user avatar
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Proving that the average number of arrival events as $\lambda t$ given the inter-arrival duration are i.i.d. Exp($\lambda$) random variables

I'm trying to prove a common result for the Poisson process but I'm stuck. Given $T_i$ are i.i.d. $Exp(\lambda)$ random variables (where $\lambda$ is the rate) that represent the duration of arrival ...
Khai Yi Chin's user avatar
2 votes
2 answers
49 views

How can you model the arrival times of 2 different events?

I was looking at a poisson process and using an exponential distribution to model arrival times of the event. For example the chances that a person is arriving at a bus stop. Then in the simple case I'...
evan54's user avatar
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3 votes
1 answer
112 views

Let $N(t)$ is a poisson process with rate $\lambda$, $T^* \sim \operatorname{Exp}(\lambda^*)$, find the expectation of $N(\min(t, T^*))$

Currently, my approach is to split $N(\min(t, T^*))$ like the following by the law of total expectation. \begin{align*} &E(N(\min(t,T^*))) = E(N(t \wedge T^*)) \\ = {}&E(N(t\wedge T^*) \...
yuw444's user avatar
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1 vote
0 answers
57 views

Poisson process for small sample size

Suppose I have a data set that contains insurance claims $X$. Each claim also is assigned the calendar year during which the insured event happened. Now, I want to build a frequency/severity model but ...
Algebro1000's user avatar
1 vote
1 answer
114 views

Simulating Nonhomogeneous Poisson Process - Conditional distribution of arrival times

For a Poisson process having rate $\lambda$. Given the number of events by time $T$ the set of event times are iid Uniform $(0,T)$ random variables. Suppose that each event are independently counted ...
J.doe's user avatar
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1 vote
1 answer
94 views

Let $N(t)$ be a Poisson process, compute $P\{N(s)=1,N(t)=2\}$ for any $0\leq s<t$

I got the answer as $\lambda^2e^{-\lambda t}s(t-s)$ using the properties like independent increment and stationary increments. But I can't seem to understand the steps in the solution of the book. For ...
A Y's user avatar
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1 answer
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Understanding a parameter in a bayesian Poisson model ($\beta$)

I would like to know the meaning or signification of the parameter $\beta$ in this Bayesian model. I have a Poisson model : $ s_{i} \mid \lambda_{i} \sim Poisson(\lambda_{i}t_{i})$ Where $\lambda_i\...
xenuti's user avatar
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1 vote
0 answers
73 views

Average time in which a product random variable becomes zero

Im looking for the optimal time in which a process should be cancelled before it results on financial losses. Say M_n=X_n*Y_n-c(n) for for n =1 to 12 which is the number of hours the process gets ...
Vacoiide's user avatar
2 votes
0 answers
38 views

A hard core spatial point process that a vehicle encounters as Poisson arrivals?

Does there exist a point process $X$ in the plane with the following two properties? $X$ is hard core. Discs of radius $h$ can be centered on the points in $X$ without overlapping. $X$ is ...
PtH's user avatar
  • 121
0 votes
1 answer
432 views

How to appropriately compare colony-forming units (CFUs)?

In microbial research, a common way to check growth rates of bacteria is by performing a dilution of the bacterial population and then plating the resulting dilution on a petri dish. After some time, ...
Piet van den Berg's user avatar
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0 answers
40 views

Distribution for a sequence of events which the first one follows a Poisson distribution

Imagine a simple website where a user can access and can click on a button that will refresh the page. On average, 50 000 requests are made to the webserver of this website in a month. We can assume ...
Vitor_figm's user avatar
2 votes
1 answer
84 views

Why can weekends cause harmony?

The following plot shows power spectra (periodograms) of a sample from $X_t \sim \operatorname{Poisson}(1)$ along with that same sample where: Weekends were set to zero Sundays were set to zero ...
Galen's user avatar
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0 votes
1 answer
444 views

How to prove the Poisson link function is a canonical link function?

So I'm a 3rd year undergraduate doing my thesisin football score models right now. In my thesis I want to include a proof of what the link function for the Poisson distribution is and why it relates ...
Nikhil Handa's user avatar
1 vote
0 answers
35 views

Radioactive decay as a Poisson process [duplicate]

I am trying to grapple with the following question as I self-study from the chapter on probability in my quantum mechanics textbook (Ballentine's Quantum Mechanics: A Modern Development). ...
EE18's user avatar
  • 203
2 votes
1 answer
169 views

Poisson Process: Probability distribution to describe time (distance) to successful event?

Given some length of time $t$ with successful events occurring in this interval at rate $\lambda$. Assume that only one successful event occurs during this interval of length $t$. Which distribution ...
Nic's user avatar
  • 21
2 votes
1 answer
110 views

Count process with standard deviation proportional to its mean

What is (is there) the count process, which has its standard deviation proportional to its mean? Note that I am not talking here about Poisson process, which has its variance proportional to mean. ...
Roger V.'s user avatar
  • 4,439
2 votes
1 answer
145 views

Likelihood for a log Gaussian Cox process (LGCP)

Suppose I have a log Gaussian Cox process (LGCP) $X$ with log intensity function $\lambda(x)=S(x)$ where $S$ follows a Gaussian process. Since LGCP still falls under the umbrella of inhomogeneous ...
xiaotomtom's user avatar
0 votes
0 answers
31 views

Poisson process vs selecting based on a probability

I want to write a function that returns an error with a rate of n% . I am confused if the following two ways are the same or what is the difference. poisson ...
MK.'s user avatar
  • 101
2 votes
0 answers
23 views

Aikaike Information Criterion for model with fixed parameter

I am trying to fit an inhomogenious, reinforced Poisson process to time series data using maximum likelihood estimation. The inhomogenious rate is $$ \lambda = \alpha \cdot f(t,\theta) \cdot (m +n).$$ ...
nikozoe's user avatar
  • 21
0 votes
2 answers
628 views

From Poisson to Erlang

If a customer arrives according to a Possion process with rate $\lambda$, how can I show that the time interval $X$ taken to receive $k$ customers is an Erlang-$k$ random variable with parameters $n$ ...
Alex_DeLarge's user avatar
0 votes
1 answer
32 views

Test for a Poisson process with a statistic related to clustering

I have a dataset of a realisation of a one-dimensional point process. I would like to test for a Poisson process with some statistics connected with clustering. I would like to provide a size effect ...
Vash's user avatar
  • 13
0 votes
1 answer
260 views

Calculating distribution of Poisson process at time t when a future value is known

Let $P$ be a Poisson point process with rate $\lambda$. If it is known that $P(t) = n$, how can we retroactively derive the conditional distribution of $P(k)$, where $k=t-s$ for $s<t$? My idea: The ...
user830529's user avatar
0 votes
0 answers
65 views

How to infer the Poisson rate parameter, given the probability of n events

Suppose I have a Poisson process. Suppose I am given the probability, $p$, that I will observer up to $n$ events. How can I subsequently calculate the implied value of the Poisson parameter? I am ...
user847663's user avatar
1 vote
0 answers
41 views

Reverse engineering and fitting Poisson distribution function?

I tried asking this on StackOverflow in the Excel/spreadsheets section and was redirected here. https://stackoverflow.com/questions/74143566/reverse-engineering-and-fitting-poisson-distribution-...
fbnhost1's user avatar
2 votes
2 answers
579 views

Is it valid to simulate a Poisson process with a sequence of Bernoulli trials?

In order to better understand some statistical concepts I generally try to run simulations to get to those results and see how the results match the theory. While reviewing the Poisson and the ...
rusiano's user avatar
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1 vote
1 answer
52 views

What an exponential distribution for a spatial poisson process answers

I use the Poisson distribution in virology where we try to answer: "What is the probability that X viruses enter a cell given a E(x)=MOI (=virus/cell)". ...
Hachiloni's user avatar
  • 113
0 votes
1 answer
49 views

Strong Renewal Assumption?

Just started a Stochastic Processes course and I am a bit confused over the Strong Renewal Assumption we make for Renewal (and Poisson?) processes. The assumption in my text goes as follows: "At ...
Persephone's user avatar

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