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0answers
9 views

Mean and variance of Cox process

Consider the (doubly-stochastic) Poisson point process with rate $ \lambda(t) = \rho e^{-t/\tau} $ where $\rho\sim\Gamma(\alpha,\beta)$ is a Gamma-distributed random variable. I require the mean ...
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0answers
16 views

A union bound on a continuous variable

Assume a continuous point process (say Poisson) in [0,t]. Assume that it is given that only three jumps occurred in [0,t], however, their exact time coordinate is unknown. In addition, it is known ...
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1answer
22 views

Binsize for Compound Poisson Gamma Distribution

Having data of a (putative) Poisson-Gamma process I am looking for a method to define the correct binsize for analysing my data. I need a certain range (!) of binsizes (for instance starting with ...
-1
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0answers
17 views

Application of Lévy–Khinchine formula

How can we express the characteristic functions of Wiener and Poisson processes by using the Lévy–Khinchine formula? I don't know how to find the characteristic functions of particular Levy ...
2
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1answer
121 views

Poisson process and the memoryless property

I understand that inter-arrival times of a Poisson process are exponentially distributed and therefore the inter-arrival times are memoryless. However, how about the waiting times of Poisson process ...
2
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0answers
58 views

Estimating parameters and latent variables in a split Poisson distribution using R and JAGS

I am trying to estimate parameters and latent variables in a split Poisson model that describes observable and unobservable counts in time assuming the split probability is $\pi$. An observable event ...
1
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2answers
57 views

Mean of Poisson distribution experimentally

Suppose I approximate a variable (events of certain type) by a Poisson distribution and I would like to write a program and measure the mean experimentally. The idea is that I have an algorithm which ...
2
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0answers
27 views

Testing if points on a line are over/under dispersed

I have a series of about 1000 points on a chromosome and I want to know if they are clumped, over-dispersed, or neither. The chromosome can be viewed as a 1-dimensional. I've looked over some ...
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1answer
45 views

Test hypothesis point process is Poisson

I have some data and I would like to test the hypothesis that they come from a homogeneous Poisson process. I can of course look at the inter event times and test if these are exponentially ...
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0answers
19 views

Poisson counting process parameter

Two quick questions: What's the maximum likelihood estimator of the parameter of an homogeneous Poisson counting process? To estimate $\lambda$ I'm currently using number of events/total time, ...
1
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1answer
54 views

A modelling question about point processes with heavy tails

I am trying to model a number of point processes for which I have data. If I choose to model each one using a (different) homogeneous Poisson process and estimate the rate using MLE then for some of ...
2
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1answer
43 views

Given time series data, how to model the frequency of someone changes his job?

I am given a time series data vector (ordered by months and years),which contains only 0s and 1s. ...
4
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0answers
37 views

Detecting changepoint in ratio of rates of two Poisson processes

I'm interested in a changepoint detection problem of the following scenario: Consider two Poisson processes for which we have the event times. I'm interested in detecting a change in the relative ...
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0answers
22 views

How to predict given Poisson regression?

I run some Poisson regressions with the following results: (with number of associations an individual belongs to as the dependent variable) ...
4
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1answer
56 views

Estimating Poisson process intensity using GLM

Suppose I want to build an explanatory model for events generated by an inhomogeneous Poisson process with unknown intensity $\lambda$. Each entry in my dataset represents the registration of an ...
0
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0answers
115 views

Fitting for a Poisson-Gaussian Mixture Distribution

First of all, I am rather new to statistics, so go easy on me. I am aware that the negative binomial distribution can be thought to arise as a result of letting the $\lambda$ parameter in a Poisson ...
6
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1answer
175 views

Total expectation theorem for Poisson processes

I have two independent Poisson processes $A$ and $B$ with arrival rates $\lambda_A$ and $\lambda_B$, respectively. Now, the expected time for the arrival of the next item for the merged process should ...
2
votes
2answers
86 views

Poisson arrival

Users arrive according to Poisson process with rate $\lambda$. If every third user is removed, then do the remaining users form a Poisson process with rate $2\lambda/3$? If every other user is ...
1
vote
1answer
211 views

How do we predict rare events?

I am working on developing an insurance risk predictive model. These models are of "rare events" like airline no-show prediction, hardware fault detection, etc. As I prepared my data set, I tried to ...
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0answers
41 views

Generalised (nonhomogeneous) Poisson process

Define a generalised Poisson process as an arrival process that begins at time 0 and that satisfies: The independence property: the number of arrivals during two non-overlapping intervals ...
2
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0answers
100 views

Non-parametric estimators for time-varying binomial proportion

I have a bunch of count data associated with time intervals (potentially overlapping and of variable lengths), say $(s_i, t_i, n_i, N_i)$ where $N_i$ is a count of the total number of events ...
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0answers
38 views

If Maria performs more observations per unit of time than Maximilien, how can he estimates the Maria's results from his own?

General problem Having a sequence of values $v_0, v_\Delta, v_{2\Delta}, \ldots, v_{N\Delta}$, which are measured every $\Delta$ units of time, usually we are interested in the prediction of the ...
2
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0answers
27 views

Mean length of time spent queueing in $M/E_2/1$ system?

Context: Consider a $M/E_2/1$ queueing system, where the customer arrival rate is $\lambda$ and the service time distribution has a gamma distribution with parameters $2$ and $\mu$, i.e. with p.d.f. ...
3
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1answer
101 views

Mean service time of a $M/E_2/1$ queueing system?

Consider a $M/E_2/1$ queueing system, where the customer arrival rate is $\lambda$ and the service time distribution has a gamma distribution with parameters $2$ and $\mu$, i.e. with p.d.f. ...
2
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0answers
89 views

Estimate of censored poisson process

I have a set of processes, each of which has number of events and the total length of time. I'm trying to model them as independent Poisson processes with there own rates. The rate of the ith process ...
2
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1answer
372 views

Maximum likelihood for number of events in Poisson process

I have a Poisson process with parameter $\lambda$ known. How do I compute the maximum-likelihood estimator for $N$, ie. the number of events over a specific time spell $T$. To repeat, I know ...
1
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1answer
36 views

Expected waiting time

The following is a worked example found in past papers of my university, but haven't been able to figure out to solve it (I have the answer, but do not understand how to get there). Any help in ...
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2answers
50 views

Poisson Process arrivals

This is a homework problem. Between 10 AM and 6 PM visitors arrive at the Tate Modern Gallery in accordance with a Poisson process at the rate of 6 per minute. Determine the probability that 10 ...
0
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2answers
111 views

Poisson and Exponential Distribution- Is the following question correct?

Accidents occur with a Poisson distribution at an average of 4 per week. i.e., $\lambda= 4$. Calculate the probability of more than 5 accidents in any one week. What is the probability that at least ...
3
votes
2answers
4k views

How to know if a data follows a Poisson Distribution in R?

I am an undergrad student and I have a project for my probability class. Basically, I have a dataset about the hurricanes that impacted my country for a series of years. In my probability Book, ...
1
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1answer
46 views

Question about identification for this parametrization

Assume I observe a poisson-process with a rate $\boldsymbol{\lambda}$. I would like to model $\boldsymbol{\lambda}$ as: $\boldsymbol{\lambda} = \boldsymbol{\pi}_1\boldsymbol{\lambda}_1 + ...
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0answers
48 views
1
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0answers
54 views

M/GI/inf queue in stationary distribution, how to get queue size distribution at the arrival times?

Suppose that we have an $M/GI/\infty$ queue, that is, we have infinitely many servers, a Poisson arrival process with rate $\lambda$ (i.e., random arrival times $0=t_0 < t_1 < t_2 < \dots ...
3
votes
1answer
169 views

Are there any alternatives to simulation for determining the distribution of number of events from two dependent non-homogeneous Poisson processes?

A "state of the art" model for the distribution of goals scored in a soccer match is that of Dixon and Robinson (1998) "A Birth Process Model for Association Football Matches" which accounts for two ...
2
votes
1answer
159 views

Testing for Poisson process

I have some discrete times of events and I would like to do a test to see if they are likely to have come from a homogeneous Poisson process. From www.stat.wmich.edu/wang/667/classnotes/pp/pp.pdf‎ I ...
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0answers
148 views

How to prove the independent and stationary increment of a poisson process?

Given a Poisson distribution with parameter $\lambda$ (basically a Poisson process), how can I prove that this Poisson process is independent and stationary increment? Or the memoryless property: ...
2
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0answers
84 views

Conditions for Poisson approximation of the superposition of non-Poisson processes

It is well known that the superposition of $N$ Poisson processes is itself a Poisson process with an intensity given by $\sum_{n=1}^{N} \lambda _{n}$. Conversely a superposition including any ...
1
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1answer
110 views

Low intensity Poisson estimation

I have a collection of Poisson processes each with an unknown $\lambda$. I would like to estimate $\lambda$ for each process. for each process I could take either the total number of event over the ...
5
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1answer
184 views

The gamma distribution and Poisson processes

I know that the gamma distribution with parameters $k$ and $\theta$ can be used as a model for the occurrence of events. The requirement on the events would be that their occurrence is random and the ...
3
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1answer
164 views

Poisson probability question

For part a I think its just the poisson probability of 40 with rate = 48. I'm stuck on b to d. Cars pass checkpoint A in accordance with a Poisson process at an average rate of 24 cars per hour. All ...
3
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1answer
191 views

Likelihood of multiple event times modeled as independent Poisson processes

I am modeling three events A, B, and C as Poisson processes with rates $\lambda_A$, $\lambda_B$, and $\lambda_C$ and I would like to calculate the likelihood of observing some data given my model. A ...
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0answers
81 views

Methodology of modelling sparse events

Let us say that we have a process that generates sequences of the following form: ...
1
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1answer
325 views

Ripley's K Function and L Function for Point Patterns

The following is a spatial point pattern: and these are the corresponding Ripley's K function and L function for this data: How are these functions interpreted?
3
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1answer
265 views

Prior for Bayesian Inference on Failure Rate in Poisson Distribution

I'm trying to derive the posterior distribution for the failure rate (lambda) of a process with poisson distribution. I have tried the use of an improper uniform distribution on lambda by letting the ...
2
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3answers
496 views

Bayesian parameter estimation of a Poisson process with change/no-change observations at irregular intervals

Consider a Poisson process with unknown parameter $\lambda$. We perform a sequence of $n$ observations at intervals $\overline{t}=t_1,\,t_2,\,\dots,\,t_n$. Each observation is a binary variable $x_i$ ...
3
votes
1answer
192 views

Poisson processes

I have two realizations of a poisson stochastic process, they are over the same space with rate $\lambda_{1}$ and $\lambda_{2}$. What is the probability that N elements in both sequences are the same, ...
4
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1answer
2k views

How to estimate Poisson process using R? (Or: how to use NHPoisson package?)

I have a database of events (i.e. a variable of dates) and associated covariates. The events are generated by the non-stationary Poisson process with parameter being an unknown (but possibly linear) ...
0
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0answers
60 views

singularity of the Poisson counting process for non-statistician

I would like to explain to non-statisticians the singularity of the Poisson counting process over others (if possible, in a simple sentence). Simply translating in non-mathematical terms its formal ...
0
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1answer
188 views

Instantaneous Event Probability in Poisson Process

In a homogeneous Poisson process with rate $\lambda$, what is the probability of observing an event in an "instant," that is, an infinitesimally small interval of length dt? I have read that the ...
3
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1answer
206 views

Expectation and confidence intervals of a Poisson process

A Poisson process has PDF $$P(X=k)=\frac{e^{-\lambda t}(\lambda t)^k}{k!}$$ I'm trying to find an expression for: $E[X | \lambda, t]$ Confidence intervals (i.e. find $\delta$ such that ...