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 process age and residual life probabilities

Let $A(t)$ and $Y(t)$ denote respectively the age and excess at $t$. Find the probability P[Y(t)>x|A(t + x)>s]) for a Poisson process. I need help on formulating this problem. Is it possible to ...
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Sampling from a Poisson Process

I am trying to simulate a one dimension correlated random walk. In this algorithm, the direction of a particle’s next step is correlated with the direction of it’s previous step. The particle’s step ...
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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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3 votes
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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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Renewal counting process with inter-arrival time gamma distribution: Model estimation

Let's start with the Poisson process: If $N_t$ is a Poisson process with parameter $\lambda$, then we know that the inter-arrival time distribution is an exponential distribution with parameter $\...
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Distribution of half-life from radioactive decay

Suppose that $X$ measures the half-life of a radioactive element, with decay rate $\lambda$ (per unit of time). Starting from a population of $N$ particle, I believe you can model the number of ...
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What is a model suitable for prediction of binary values based on time series?

There is a time series of stock quotes, length $n=500$. The data looks like: ...
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Proving the expectation of a variable in a stochastic process

Problem Information packets arrive at a server with a poisson process having rate $\lambda = 2$ per hour. The server processing time for a packet follows the distribution : $f(x) = 1, 0\leq x\leq1$ ...
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How to use Poisson statistics to estimate error bars on rates

I want to know the fraction of bananas which are ripe at some moment in time. If I have 100 bananas, and 25 of them are ripe, then my rate is 25%. But if I want error bars on this so I can generalize –...
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Can an inhomogeneous Poisson process $N(t)$ be obtained from a profile of the total instantaneous population size?

Suppose I have the total instantaneous population size as function of time $t$, $n(t)$. I want to obtain the sample arrival process $N(t)$ (a counting process, which is non-decreasing), assuming that ...
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Generating random data points following a Poisson point process from observed data points [duplicate]

I am a bit new to the domain of Spatial Statistics. So I am trying to generate complete spatial randomness(CSR) data with summary statistics similar to that of the observed data(data points in 2D). ...
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Renewal process: Stationary and independent increments?

For a renewal process, we know that the inter-arrivals are independent but not exponentially distributed, as opposed to the Poisson process for which the inter-arrivals are exponential. We also know ...
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Deriving the Newton-Raphson update for an MLE with two parameters

Suppose we use the Poisson process assumption given by Ni|bi1,bi2 ~ Poisson(λi) where λi = α1bi1 + α2bi2. The parameters of this model are α1 + α2 which represent rate. How do we derive the Newton-...
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Assumptions of compound Poisson model

My understanding of a compound Poisson RV is one defined as $$Y=\sum_{n=1}^N X_n$$ where $\{X_n\}_{n\in\mathbb{N}}$ is a sequence of identically distributed and mutually independent (iid) RVs $N$ is ...
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Proportional Poisson Process Problem

The alliteration is unintentional, but I was amused so I'll leave the title as is. Anyways, I am looking at a problem where we are trying to predict/justify an observed instantaneous rate by which ...
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Cox process (doubly stochastic NHHP) simulation

I want to simulate a Cox process to model demand for booking requests at an airline, where the arrival intensity at time $t$ has gamma distribution (see Weatherford, L. R., Bodily, S. E., & ...
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Computing relative likelihoods of sequences relative to Poisson process

I have a problem where I need to compute the likelihood of a given sequence of events (in time) given a Poisson process with a particular lambda (events per second). I only need it to within a ...
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How to implement the Kolmogorov forward equation for a Poisson process?

How could I implement the Kolmogorov forward equation on the Poisson process? I know that Kolmogorov forward equation is: $$ P_{ij}'=\sum_{k \neq j}(λ_{k} P_{kj}P_{ik}(t)) - λ_{j} P_{ij} $$
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Measuring cumulative number of events in a day

A certain type of event happens in continuous time throughout the day. During some periods of the day (say between 9 am and 5 pm) the event happens more frequently than outside of these times. Given ...
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Step in proof of poisson probability

for the case when $s < t$ Let $N(t)$ be the amount of arrivals occurring at time period t in a Poisson process. When $N(t)$ is known, the number of events by time $s$ can then be shown to follow a ...
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95 percentile for failure frequency with Poisson process

I am trying to recreate the stats for 'failure frequency' - in this case the 95 percentile - shown in the attached images below. For one set of failure data there are 27 failures over a cumulative ...
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1 answer
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How to calculate the confidence interval of a mean on web search findability

I have 30x 1-hour sessions where I assign one random person to each session to search for a specific type of content on a social media website. I record the number of pieces found in each session, ...
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Insurance Claims: Proving a Process is a Poisson Process and Finding its Rate

Let $X(t)$ denote the number of claims received by an insurance company in the time interval $[0,t]$. We will assume that ${X(t) : t ≥ 0}$ can be modelled as a Poisson process, where $t$ is measured ...
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Adding a lognormal distribution into a model of shot noise for a Poisson process to calculate moments of the process

I'm working with a simple 1D model of some objects propagating through a domain, $x$. The setup is like shot noise, and is actually taken from a paper by Militello, F., and Omotani, J. T, 2016 Nucl. ...
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Time Series vs. Queueing Models

Generally speaking, queues are modelled using the Poisson Process. Supposedly, this used to model the dynamic nature of queues, arrivals, birth-death and renewals. But just as a basic question: Why ...
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Expected value of conditional Poisson process

I have been trying to solve this issue for quite a while. So, lets say that we have a Poisson process, $N = (N_t, t \geq 0)$ and the $\lambda = 3$. Lets say that $Y = (N_2 | N_6 = 3)$. Find $Ee^Y$. ...
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3 votes
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What kind of physical processes are well modeled as poisson processes?

For instance, it is easy to understand how Brownian motions appear: when an object receives many small shocks that are roughly i.i.d., the resulting path will look like Brownian motions. What type of ...
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Can a time dependent lagged function be used as a rate parameter?

Question: Can the rate parameter in a mm1 queue or a Poisson process be a lagged function of itself? https://en.m.wikipedia.org/wiki/Poisson_point_process https://en.m.wikipedia.org/wiki/M/M/1_queue ...
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Queues with non-constant arrival rates

https://cran.r-project.org/web/packages/simmer/index.html I am interested in simulating some basic queues in R. However, I want to try to simulate queues that have "non-constant" arrival ...
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Can any discrete stochastic be considered as a "point process"?

"A point process is a stochastic model underlying the occurrence of events in time and/or space. In this blog, we will emphasis on purely temporal aspects of point process i.e., the space in ...
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Statistical Test for Independence Increments

https://www.youtube.com/watch?v=aCSocDPw8cM @7:45 - They mention in this video that for a Counting Process to be considered as a Poisson Process, it must at least have "Independent Increments&...
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MLE for $\lambda$ on the Poisson process using exponential inter-arrivals

Context: The Poisson process Take $m$ independent and identical Poisson processes with rate parameter, $\lambda$ running in parallel (we can assume they start at the same time, but shouldn't matter ...
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Minimum variance unbiased estimator for the parameter, $\lambda$ of a Poisson process

Consider a Poisson process. We start observing it at a time, $u_1$ in its time-line, until the time $u_2 = u_1 + u$. There are $m$ of these processes operating independently of each other. This is ...
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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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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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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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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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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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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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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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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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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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3 votes
1 answer
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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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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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1 vote
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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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