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1answer
36 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
21 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 ...
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0answers
71 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
28 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 ...
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0answers
22 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. ...
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1answer
68 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
47 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
175 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 ...
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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 ...
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2answers
43 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 ...
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2answers
73 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
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2answers
869 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, ...
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1answer
37 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
39 views
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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
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0answers
36 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 ...
2
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1answer
102 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
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1answer
91 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
113 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
76 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
96 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
132 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
154 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
116 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
66 views

Methodology of modelling sparse events

Let us say that we have a process that generates sequences of the following form: ...
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1answer
217 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
194 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 ...
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3answers
360 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
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1answer
153 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, ...
3
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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) ...
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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
118 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
187 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 ...
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0answers
98 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
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1answer
123 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: ...
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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
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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". ...
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1answer
299 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
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1answer
1k 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
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1answer
214 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
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1answer
146 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
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1answer
176 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 ...
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6answers
1k views

Is there any gold standard for modeling irregularly spaced time series?

In field of economics (I think) we have ARIMA and GARCH for regularly spaced time series and Poisson, Hawkes for modeling point processes, so how about attempts for modeling irregularly (unevenly) ...
2
votes
3answers
499 views

Can I estimate the parameter of a Poisson arrival process from a low-incidence observation period?

If I know only that the arrival process is Poisson, and I observe it for a pre-chosen (say, unit) period of time, observing $k$ arrivals, is it meaningful to describe an estimate of its time parameter ...
1
vote
1answer
264 views

Estimating event duration by sampling with a Poisson process

Take an event to be a set $E \subseteq {\mathbb R}^{\ge 0}$. We want to estimate the duration of the event (the measure of $E$) by sampling using a Poisson process $N$. That is, for some experiment ...
5
votes
1answer
195 views

Number of events of type B given n events of type A in a Poisson process

I have a simple probability calculation I was working on that I came up with an answer for, but a question a colleague asked me led me to come up with a second approach - and a different answer. I put ...