# 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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### Confusion about Poisson rectangular pulse model

I am reading a paper by Rodriguez-Iturbe et al. from 1986 and am confused by the below derivation. The model they are working with is a Poisson process with rate $\lambda$ in which each occurrence in ...
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### How to calculate the average inter-arrival time from a series of inter-arrival time?

I am currently handling count data and just look into the poisson process. It maybe a trivial question but I am really confused on it. I have collected count data and the inter-arrival time between ...
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### what are the parameters of a poisson term in a jump diffussion SDE

I am trying to simulate a Jump diffusion process in python and not sure what i should specify as the parameters of the poisson term. Here is what i have ...
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### Can we consider arrival of pest outbreaks reports as Poisson Process?

I have a scenario where given a geographical region, farmers within that region can generate pest outbreak reports (like twitter messages) using a mobile phone. Also, a particular pest can spread to ...
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### Finding the probability of survival of an insurance company

I was given as a homework exercise the following problem: however, I came into a disagreement with one of my classmates. Given that the solution is not shown, I was wondering whether mine was correct....
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### Poisson Process conditional probability

Let $N(t)$ be a Poisson Process with rate $\lambda$ Find $\displaystyle P(N(4) \le 2N(2) \mid N(2) = 1) = \frac{\sum_{i = 0 }^2P(N(4) = i, N(2) = 1)}{P(N(2) = 1)} = ?$ Can I split this up using the ...
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### How can I predict the total number of cell with Poisson process

I detected cells in circulation with our own system, the cell is very rare compared to other blood cells, I acquired the temporal information of each cell and calculated intervals between 2 cells. The ...
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### Multivariate Bayesian Car Model Result

I have developed a multivariate Bayesian Car model for three crash severity level analysis. I found that the covariance for both heterogenous effects and the spatial effect is not significant for any ...
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### Confidence interval over multiple Poisson distributions

There are five different periods (one following each other) of unequal lengths where one specific event is happening such that it follows Poisson distributions (e.g. meteorite falls). It is always the ...
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### Probability of a 500 year flood occuring in the next 100 years - comparison of approaches

I'm looking at this problem A $500$-year flood is one that occurs once in every $500$ years. a) What is the probability of having at least $3$ such floods in $500$ years? b) What is the ...
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### Poisson process as a spatial process

Let's consider a Poisson process on the line with rate parameter $\lambda$. There are two ways to think about this: In any interval $[a,b)$ the expected number of events is distributed as a Poisson ...
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### Simultaneous Poisson processes

The arrival of taxis at a taxi stand is Poisson at rate $\lambda$ per hour. The arrival of people to the stand is also Poisson but at rate $\mu$ per hour. Taxis do not stop (they leave empty) if ...
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### Pareto/NBD with time-varying covariates

I am trying to incorporate time-varying covariates into the Pareto/NBD model (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.597.3165&rep=rep1&type=pdf) Model assumptions start at ...
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### Does it make sense to infer a rate (as a probability distribution or upper limits) for a Poisson process if there are “no events”

I have an inhomogeneous Poisson process with a rate $\lambda (\mathbf{t})$ defined on some parameters $\mathbf{t}$. I am trying to infer $\lambda (\mathbf{t})$ from some data, which are events (really ...
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### Smallest gap between $n$ Poisson events

Given a Poisson process with parameter $\lambda,$ run time forward until the first $n>1$ events occur. What is the distribution of the smallest time interval between two events? This is surely ...
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### MLE estimation of poisson intensity for a normal compounded poisson process

I was working on a project where a time series is modeled as compound poisson distribution with normally distributed jump levels and I want to get MLE estimator for the poisson intensity, jump mean, ...
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### Poisson regression for count data that is not Poisson distributed?

Is it true that Poisson regression is used to model count data? But not all count data follow a Poisson distribution? Then you can still use Poisson regression in that case?
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### Comparison of Poisson cumulative probabilities between models with different rates

I'm building an anomaly detection model using the Poisson distribution. I'm calculating the Cumulative Probability: P(X >= x) for thousands of count data, all of them with different rates. I need to ...
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### Alternating between two states {A, B} each with exp distributed durations. What's the probability of state=A at time t?

Say I have a light bulb that can be on (A) or off (B). It alternates between being state A or B. It will be in state A for a duration a ~ exp(α), and in state B for duration b ~ exp(β), (...
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### Conditional distribution of arrival times in Poisson process

Suppose I know over a window $[0, T)$ that I have observed $n$ samples from a poisson process $N_t \sim p(n|\lambda t) = \frac{1}{n!}(\lambda t)^{n}\exp(-\lambda t)$. What is the conditional ...
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### Poisson Process With Two Different Rates

I am looking to investigate a Poisson Process that has two separate rates; essentially, there is a season of games that each have their own points rate (per minute). Comprehensively, over the season, ...
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### Overdispersion under different longitudinal constraints

I have panel data (police districts observed across many months) and I am modeling a count outcome. Fitting a Poisson model (simple pre/post comparison with 6 months of data) results in significantly ...
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### How to compute the expected number of events in the following conditional renewal process?

I have a stochastic point process with event times $\{x_1, x_2, ...\}$ and I want to compute the expected number of events $n(T)$ over the interval $[0,T]$. The point process is generated as follows: ...
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### How to show the variance of the inter-arrival time of a Cox process driven by a Poisson process of constant intensity $\lambda$ is $3\lambda$

Ideas on how to show that the variance of a doubly-stochastic Poisson process(aka a Cox process) driven by a homogeneous(stationary) Poisson process of intensity $\lambda$ is $3\lambda$ ? I've come ...
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### To detect which of the three processes more-aptly fits Poisson process

There happens to be a type of generic poll, which can be described as:- A participant can vote only once and for only 1 of 3 running candidates - Red, Blue and Green. A random voter starts the ...
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### Solving an equation with logarithms of negative numbers

I'm modelling a bunch of non-homogeneous poisson processes. I'm impossing on them a linear-log functional form. you end up with two parameters, theta1 and theta0 for each estimated process. The ...