# 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 ...
1 vote
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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 ...
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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 ...
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### 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 ...
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### 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 ...
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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 ...
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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 ...
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### 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 ...
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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 ...
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1 vote
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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 ...
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### 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 $Y=(Y_1, Y_2, \dotso, Y_M)$ ...
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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 ...
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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 $N=N_{1}+N_{2}$ ...
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### 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 ...
1 vote
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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 ...
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### 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 ...
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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 ...
1 vote
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### 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 ...
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### 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 ...
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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 ...
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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 ...
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### 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'...
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### 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^*) \...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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-...
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### 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 ...
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