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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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568 views

Calculate conditional probability for the Poisson Process

I need to calculate $P[N_{s}=k||N_{u},\,u \geq t]\,(s \leq t)$ for the Poisson process. However, I have been instructed to use the following example in order to do so: Example: The Poisson process $...
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1answer
484 views

Histogram for a compound poisson process

Would a compound poisson process result in a histogram that isn't the same as that of a regular poisson process? How would I fit such a histogram without knowing the rate?
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206 views

Inter-arrival time of Poisson arrivals?

Assume a Poisson process with rate $\lambda$. Let $T_{1}$,$T_{2}$,$T_{3}$,.... be the time until the first, second, third,......(so on) arrivals following exponential distribution. If I consider ...
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1answer
80 views

Derivation of probability under assumption of Poisson process

Poisson process start with certain assumption about the how process govern in short interval of time $\Delta t$. The first assumption about the Poisson process is that the probability of occurrence ...
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1answer
395 views

Compound Poisson Process with Weibull jumps

I need to simulate a compound Poisson Process in R, however I am not clear with the algorithm to generate it. I have conceptual gaps. I know by definition that: A compound Poisson process is the ...
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0answers
269 views

zero inflated poisson model, how to choose “inflate” variables

I am trying to run a zero-inflated Poisson regression. The data I have are number of West Nile Virus cases NoCases (dependent var), and my independent variables are AvgTemp, AvgPrecipitaion, Region (*...
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1answer
331 views

Poisson Process Intensity Function

I'm working through Kingman's Poisson Processes and have a question about defining the mean measure. Why is it that in most cases we specify an intensity function $\lambda(x)$, where the mean measure ...
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2answers
68 views

How to quantify the reliablility of an estimate

Lets say I inspect two workers. Worker A works one hour and produces 2 goods. Worker B works 100 hours and produces 180 goods. So, on average, worker A produces 2 goods per hour and worker B 1.8 ...
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2answers
3k views

Simulation of a Poisson Process

I am trying to simulate the compound Poisson process using the next algorithm that I found in a textbook on stochastic processes. Let $S_0 = 0$. Generate i.i.d. exponential random ...
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1answer
23 views

Poisson: Probability it will take less than x time for more than y to occur

I am able to answer a poisson question where say: expected value = 2 per week, whats the probability greater than 7 in 3 weeks. But how would I answer a question where, for example expected value = ...
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44 views

Poisson distribution inside-out: estimate the beginning of the period based on first occurrence time

If a Fisherman catches his first fish of the day at 10am, when is he more likely to have started fishing given that in average he catches 1 fish/hour? In other words: Let's assume a Poisson process ...
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716 views

Equation for Inverse Poisson CDF

I am attempting to calculate quantile probabilities. I.e., the value above which there is only a 1% chance occurrence for an arrival process. The R code is pretty straight forward with say a lambda = ...
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114 views

How should I smooth / average semi-sparse service usage data used in a time series regression?

I am using time series regression and survival models to predict future user outcomes based on past usage trends. I am dealing with a data set of daily service usage per user which, on a given day ...
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2answers
599 views

Poisson process: getting a poisson from an exponential assumption

In a poisson process, getting an exponential distribution from a poisson assumption (for eg. in a given period of time $t$, if number of books that arrive is given by a Poisson, then the time until ...
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0answers
200 views

Poisson Distribution Process Control Limits

I am measuring the number of events that happen within n minute intervals for a group of 100 stores. My problem is, I am looking for a way to calculate a threshold for the acceptable number of times ...
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2answers
247 views

Motivation for gamma distribution with a non-integer parameter

The Erlang distribution has a straightforward interpretation in terms of waiting time for the occurrence of a predefined number of events in a Poisson process or a sum of a predefined number of ...
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1answer
103 views

Testing Poisson process where $X(t)$ is given at fixed times

I have a discrete stochastic process $X(t)$ which I believe is a Poisson process, that is the value of $X(t)$ at time $t$ is a Poisson random variable with parameter $\lambda t$ and disjoint intervals ...
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1answer
405 views

Is it valid to simulate a shot noise Poisson process with discretization (R)

I have a two-factor shot noise process as follows (sorry for the picture of text): I want to simulate from this process. Is it valid to do so in a discretized way? The solution to these SDEs is a ...
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37 views

Homogeneous poisson process and identical distribution

Suppose that the arrival of men and women at a bank can be considered independent homogeneous Poisson processes with mean 10 and 8 at every 30 minutes, respectively. In a interval of 10 minutes, what ...
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2answers
55 views

Why isn't there a bound on the waiting time for the first occurrence in Poisson distribution?

From my book: Let $W$ denote the waiting time until the first occurrence during the observation of a Poisson process in which the mean number of occurrences in the unit interval is $\lambda$. ...
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1answer
212 views

Simulate Homogeneous Poisson where each event is uniform

The problem is the following: Buses arrives according to a Homogeneous Poisson with arrival tax of 5 per hour. Each bus can contain 20, 21, 22... 40 passengers with eaqual probability. ...
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5answers
5k views

How to simulate Poisson arrival times if the rate varies with time?

Suppose we are to study a non-homogeneous Poisson process of 3 hour cycles in which: At the first hour, the arrival rate is 1.5 events / hr. At the second hour, the arrival rate is 2.1 events /...
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2answers
94 views

Any expression for the probability of a hard sphere in Boolean model

I am working hard on a problem of Boolean model. In a example of Boolean model, points are scattered in the plane according to a homogeneous Poisson process of intensity λ. On each of these points a ...
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1answer
261 views

Time Rescaling Theorem and Residual Analysis

Let $\mathcal{P}$ an homogeneous unit rate Poisson process. It's conditional intensity function (star indicating conditioning on the history) can be written as $$\lambda^*(t) = \lambda = 1$$ meaning ...
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0answers
163 views

Howto derive statistical upper limit in case of zero observation of poisson process?

I am writing software, that keeps track of hourly rates of certain incidents in historic data. I find that the generating process of these incidents is acceptably well described by a poisson process. ...
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3answers
365 views

examples on sequence of poisson random variables

Let $X_n$ be distributed as a poisson random variable with parameter $n$. Then which of the following are true ? 1.$\underset{n\rightarrow \infty}{\lim} \mathbb{P} (X_n > n + \sqrt n)=0 $ ...
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1answer
3k views

Likelihood Ratio Test for Poisson Distribution

Suppose we have the following count data: ...
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0answers
53 views

Calculating the p-value of two independent counts? [duplicate]

My simple experiment: Two equally-sized patches of the night sky are observed: Patch A contains $100$ stars Patch B contains $110$ stars My null hypothesis is that stars are randomly distributed in ...
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1answer
243 views

Probability of an independent Poisson process overtaking another

I have asked this question before in another fashion on other stackexchanges, so sorry for the somewhat repost. I have asked my professor and a couple of PhD students about, without a definitive ...
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0answers
259 views

Probability of a random occurrence of Poisson process to be the first occurrence

It is given, that there is an occurrence of Poisson process (at time $t$) of intensity $\lambda$ within an interval $(0, T)$. $t \in \{t_i\}; 0 < t_1 < t_2 < \cdots < t_n < T$ I need ...
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0answers
28 views

Total hourly profit for a single-server food stand

Customers arrive to a single-server food stand according to a Poisson process with rate $20$ per hour. The time to serve a customer is exponentially distributed with a mean of $2$ minutes. (a) The ...
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2answers
188 views

Crossing a road through a Poisson process

Am currently working on a Stochastic Poisson process on my project. I have thought and settled on the below scenario which I think is appropriate. However, solving it am not getting what I expect. I ...
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3answers
380 views

Paradox of Poisson process with at least one event in the interval

Let $X_T$ is a number of events in Poisson process of unitary rate ($\lambda = 1$) within interval of length $T$. It is known that at least one event has been observed in the interval, I want to find ...
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138 views

Evaluation metrics for cluster or cox process

I am working with spatial point processes and on a dataset which seems to be a non-homogeneous poisson point process. I have fitted a cluster/ cox process model and also used this model to predict the ...
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1answer
2k views

Distribution of inter arrival times in a Poisson process

I am new to Statistics. I am studying Poisson process, I have certain questions to ask. A process of arrival times in continuous time is called a Poisson process of rate $\lambda$ if the following ...
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37 views

Expected ratio of right-censored to non-censored waiting times in Poisson proces

I am trying to calculate the expected value of $Y = \frac{\sum_{i=1}^n min(x, X_i)}{\sum_{i=1}^n X_i}$, where $X_i$ are independent random variables of exponential distribution of ratio $\lambda$. I ...
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111 views

Poisson and Negative Binomial Models - Are real counts really necessary?

The Poisson model is probably the first that pops into mind when trying to model count data, which is generally described as: non-negative integer data. The Poisson distribution is defined as modeling ...
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136 views

Test if one Poisson process's rate is smaller than a given value

Suppose $N(t)$ is a Poisson process with rates $\lambda$. Suppose I've been observing it for $t \in [0, T]$ and recorded events. How can I test the null hypothesis $\lambda < \lambda_0$, where $\...
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177 views

Interarrival times of Negative Binomial (Yule-Furry) process?

The interarrival (ia) times of a Poisson process are exponentially distributed. What distribution function describes the ia times of a Negative Binomial process aka Yule-Furry process? Thanks!
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1answer
193 views

NIST exponential distribution Poisson distribution

An exponential distribution describes the time between events in a Poisson process. Suppose that the average waiting time for an action is 5 minutes. The time waited each time measured in hours is an ...
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1answer
144 views

Alternatives to Poisson Distribution

Imagine a data set that has information about how frequently various people visit a location $l$ over a year ($l$ could be a restaurant or a public park). What would be a good way to analyze this data?...
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147 views

Integral of a continuous family of i.i.d bernoulli random variables

Let $F(t) \sim \operatorname{Ber}(p)$ for every $t \in [0,1]$. Let $X = \int_0^1 F(t)\,dt$. $X$ is of course itself a random variable. Questions: Does $X$ exist? If so, what is the distribution of $...
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121 views

Memoryless Property

Here is question I am working on to study for an exam that I am quite not sure how to frame a proper answer for: Trains arrive at a station with i.i.d interarrival times following an exponential ...
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2answers
109 views

Poisson probability of more than 200 events

The $N(t)$ is a poisson process for the number of events to occur with a mean $\lambda(t) = 3$ per day. I am supposed to find the probability of more than 200 events in 60 days. My theoretical answer ...
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2answers
482 views

Poisson distribution problem - traffic problem

Hi So I have this question below. I know I need to model each lane as a separate Poisson distribution. The possible answers are: a) 11.4%; 22.4%; 33.4%; 44.4%; 55.4% b) 2.74%; 4.74%; 12.74%; 34.74%; ...
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1answer
795 views

Compound Poisson random variable

A compound Poisson random variable $S$ is defined as: $S=\displaystyle\sum^N_{i=1}X_i,$ where $N$ is a random draw from a Poisson distribution with intensity parameter $\lambda$, and $X_i$ are ...
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49 views

How do I evaluate the accumulated error of a repeated Poisson process?

I'm quite newbie in stats, so sorry if I write anything stupid. I'm dealing with a statistical process in my master in electronics and I'm kind of stuck. Consider a device that counts individual ...
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110 views

Burke's Theorem for rejection from Erlang-B loss queue

I have some general uncertainties regarding the rejection process from an Erlang-B loss queue ($M/M/c/c$), where the total capacity of the queue is equal to the number $c$ of servers. Consider the ...
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59 views

Marking a hard core poisson process

A hard core process (HCP) deals with the deposition of hard spheres, generally of the same radius, that are forbidden to overlap. Suppose instead that the identical spheres are replaced with distinct ...
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131 views

Proof for the distribution of the increments of Gaussian and Poisson processes?

What is analytic proof for the fact that increments of a Gaussian process are again Gaussian? How could I apply that to increments of Poisson process? Maybe do I multiply characteristic function ...