# 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 ...
1 vote
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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:...
1 vote
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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: ...
1 vote
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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 : $f(x) = 1, 0\leq x\leq1$ ...
1 vote
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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-...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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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### 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 ...
1 vote
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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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### 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. ...
1 vote
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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 ...
1 vote
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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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### 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 ...
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 ...