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2
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1answer
36 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. ...
0
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0answers
26 views

Independent Poisson processes

I think I read once that two Poisson processes are independent if and only if the expected difference of their log-intensities is zero. Is there a nice proof or better yet a citation?
4
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0answers
23 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 ...
0
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0answers
19 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
votes
1answer
41 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
57 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
votes
1answer
118 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
85 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
98 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 ...
1
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0answers
32 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
87 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 ...
0
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0answers
34 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
votes
1answer
86 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
votes
0answers
67 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
votes
1answer
237 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
vote
1answer
33 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 ...
1
vote
2answers
47 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
votes
2answers
95 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 ...
2
votes
2answers
2k 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, ...
0
votes
1answer
40 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
43 views
0
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0answers
26 views

How to model ties between two event sequences?

Context I have two processes that each emit an event at various times: ...
1
vote
0answers
50 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
141 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
132 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 ...
0
votes
0answers
125 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
votes
0answers
81 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
vote
1answer
108 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
votes
1answer
151 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
votes
1answer
161 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
votes
1answer
161 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 ...
1
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0answers
72 views

Methodology of modelling sparse events

Let us say that we have a process that generates sequences of the following form: ...
1
vote
1answer
281 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
votes
1answer
220 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
votes
3answers
420 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
170 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
votes
1answer
1k 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
votes
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
votes
1answer
148 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
votes
1answer
197 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 ...
1
vote
0answers
103 views

Compound poisson process: Average size of claim will exceed £110

"An insurance company receives claims at a rate of two per week, the size of a claim in pounds having mean 100 and standard deviation 50. Assuming the compound Poisson process as a model, and using ...
3
votes
1answer
138 views

Regression for poisson process in R

I have a series of samples of varying length, and the number of bugs created in those time samples. Reading the literature, this is often modeled as a Poisson process. If you write it like: ...
2
votes
1answer
98 views

Finding the PMF of conditional probability, poisson process. Don't understand where $10^6$ goes

"Customers arrive at a bank according to a Poisson process with rate 6 per hour. State (together with a proof) clearly the (conditional) probability mass function of the numbers of customers arrived ...
2
votes
1answer
87 views

Let $\{N(t), t \geq 0 \}$ be a $PP(\lambda)$. Compute $P(N(t) = k | N(t + s) = k + m)$

The question is: " Let $ \{N(t) , t \geq 0 \} $ be a $PP(\lambda)$. Compute $$P(N(t) = k | N(t = s) = k + m), $$ where k and m are non-negative integers and $ t, s \geq 0 $ are any real numbers". ...
4
votes
1answer
323 views

Distribution of arrival times to server for an M/M/1 queue (what the server experiences)

In an M/M/1 queue, we know that inter-arrival times are exponentially distributed, and that service times are the same. What is the distribution of to-server inter-arrival times (aka service start ...
5
votes
1answer
2k views

What are the differences between survival analysis and Poisson regression?

I'm working on a classical churn prediction problem using the number of visits of a given user to a site and I thought that Poisson Regression was the right tool for modelling the future engagement of ...
2
votes
1answer
227 views

Poisson process thinning females and males arriving

Rock tickets are sold at a ticket counter. Females and males arrive at times of independent Poisson processes with rates 30 and 20. What is the probability that the first three customers are ...
1
vote
1answer
167 views

No-simultaneous-events assumption of the Poisson process

I am checking the description of Poisson process from Wikipedia. The Poisson process is based on four assumptions, but I am not clear this: No counted occurrences are simultaneous. Can you ...
5
votes
1answer
185 views

What are finite window effects?

I'm reading a paper that uses a Poisson process to model real world events. The authors mention "finite window effects". What are finite window effects? Here is quote from the paper where the ...