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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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 ...
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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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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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Tail of the maximum of a time-varying Poisson-GP marked process
Consider a time-varying version of the Poisson-GP marked process on
the real line as commonly used in Peak Over Threshold (POT) modelling
of a variable $Y$. More precisely we have a given time-varying
...
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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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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\...
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Multiple mean comparison test for a Non homogeneous Poisson Process
I do statistics for a restaurant in which people arrive at random times and end up paying a random amount of money when they are finished. I am using a Compound Poisson Process to model this, but I ...
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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 ...
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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 ...
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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, ...
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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 ...
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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
...
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Main event time prediction based on different sub events
As the title says, I want to predict the time (with a wide error range) of a main event’s first occurrence based on previous sub events that are vary in importance. These previous ‘predictor’ events ...
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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 ...
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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). ...
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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 ...
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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. ...
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Conditional probability for Poisson process
Give this question:
For a Poisson process with rate λ, find P(N(s) = k|N(t) = n) when s > t.
What is the difference if it was given that s < t?
If s > t, do the two events become ...
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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 ...
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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 ...
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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
$$ \lambda = \alpha \cdot f(t,\theta) \cdot (m +n).$$
...
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Parameter estimation of a 'parametric' Poisson process
I observe a sample of the form $S = [S_1, S_2, \ldots, S_N]=[(t_1, x_1), (t_2, x_2), \ldots, (t_N, x_N)]$ where each $t_i$ is an arrival time and $x_i$ is the amount of money spent by the $i$th ...
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I want to know if age and residual life time of the Poisson process are independent
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
$A_t := t-T_{N(t)} $,the $...
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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 $n$ ...
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Poisson process and point process
I have difficulties in understanding point process.
Here is my problem : We start with the following definition of a point process
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and $(...
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Tandem system in queuing theory
We have a queue networking comprising of two queues $A$ and $B$ such that customers after finishing the service in $A$ go directly to $B$ then from $B$ go out of the network after service in $B$. ...
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Compute adjustment coeficient for Poisson process with uniformly-distributed claim sizes
I have to aproximate the ruin probability using $ \psi(u) \approx e^{-Ru} $. The claim sizes follow a uniform distribution with mean $\mu=200000$ and standard deviation $\sigma=50000$. The claims ...
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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 ...
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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 ...
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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 ...
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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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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)".
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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 ...
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Random walk and Poisson process
(1) A point is chosen at random in a circle with center at the origin and radius R. That point is taken as the center of a circle with radius X where X is a random variable having density f. Find the ...
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Should we complete missing combinations with 0 in count data?
I'm working on count data preparation which will be used in Poisson/GLM. Specifically for year 2002 both males and females rows are missing as no event has been recorded (0 counts). Therefore should I ...
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Is it possible to estimate the parameters of a superposition of Poisson processes through Bayesian inference from a binarized sequence?
My question is complementary to a previous problem : Bayesian inference on binarized Poisson distribution. I retake the previous notations. Problem description :
I am counting the number of balls ...
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Variance and Mean Relationship from Simulated Poisson Process with Sampling
Background: I have a simple simulation of a sampled thresholded Poisson Process that arrives at a closed form solution but need help with the proof.
In my example, I am simulating 10,000 silicon ...
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Does the arrival of vehicle on a specified point of a road follow a poisson process?
I am asking about a poisson with the same rate but different serving time, because a point in a road have a maximum capacity of letting a number of cars pass through it, but in the same time the speed ...
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Poisson processes: Joint probability in overlapping intervals
Two teams, A and B, play a soccer match. The number of goals scored by Team A is modelled by a Poisson process $X_t$ with rate $\lambda = 0.03$ goals per minute. The number of goals scored by Team B ...
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Seasonality test for poissonian counts
I have a count $N_i$ of rare evevents for each season, where $i$ identifies the season. The counts' rates are not high enough to justify a normal distribution aproximation of the poissonian.
I want to ...
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Confusion on units for the Poisson distribution when it is used to model variables with units
This question stems from the comment section of this question: Bus wait time under Poisson distribution, where it seems that
The properties of the Poisson don't make sense for times because the units ...
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Bus wait time under Poisson distribution
A bus will depart every 10 minutes from the origin, and the time it takes to travel to station $A$ follows a Poisson distribution with expectation of 10 minutes.
Alice arrives at station $A$ around 9:...
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Algorithm for generating a Poisson process on a complicated 2d geometry
I am looking at some count data by geographic counties in California. As a starting point, a Poisson process came to mind--though there are other good choices like negative binomial, etc.
Given a $\...
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