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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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Data of daily counts modelled as Poisson process: should it be compound?

I understand that one of the basic assumptions of a Poisson process is that in a small enough interval, the probability of more than one arrival is negligible. In my case, I have data that shows the ...
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107 views

What to do with zero-counts when fitting Poisson distributions? [closed]

Suppose we have events that occur in time following some compound Poisson process. For example, let's say the rate parameter is such that an event occurs every 30 days, on average. If we have 10 years ...
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319 views

MLE for a homogeneous Poisson process?

If we have a data set consisting of event times $\{t_1, t_2, \ldots, t_N\}$ and would like to model this as a Poisson process with intensity $\lambda$, how do we do it? Intuitively, I would expect ...
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Parameter estimation for NHPP for arrival series

I have multiple data sets (about 15) that describe a process of arriving customers to a shop. When plotted it's pretty clear that over time the rate of customers arriving decreases. However the time ...
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39 views

Poisson processes or regression for lambda

I am asked to solve a problem where I have a machine producing toys in tho slots and want to predict number of faulty toys. The data is like this: ...
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221 views

What is mean by the term “constant rate” in Poisson distribution?

I have difficulty in understanding the assumptions of Poisson distribution, one assumption is the rate at which the events occur in the time interval is constant. What is the meaning of that phrase? ...
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Compute the covariance of random variables in a discrete-time queueing process

Background Suppose we have $Z$ particles at time $t=0$, with $Z\sim\text{Poisson}(\lambda)$. At each time $t=0,1,\dots,$ a particle either expires or continues to exist. Suppose that for each $t=0,1,\...
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Find unbiased estimators for $\lambda$ and $\lambda^2$.

For the spatial homogeneous Poisson process, find unbiased estimators for $\lambda$ and $\lambda^2$. Attempt: Since the homogeneous Poisson process is over an area, how i would i go about ...
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599 views

Prove that the interarrival times of a Poisson Process are all indipendent and identically distributed

{$N_t$} with $t\in \mathbb{R}$ is a Poisson process with intensity $\lambda \in \mathbb{R^+}$, so that 1) $N(0)=0$, 2) {N(t) is with indipendent increments and omogeneous increments and 3) $\...
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Help me understand poisson.test?

I want to understand the poisson.test() function: ...
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87 views

How do I compare two event rates?

Say I have two event streams, modeled as Poisson processes. These streams omit events at rates $\lambda_1$ and $\lambda_2$ respectively, so the count of observed events in $t$ time is $\lambda_1t$ and ...
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27 views

Compound Poisson process for demand

I have a demand pattern for a service part. Demand event rate of this part is Poisson distributed. Demand of the part is 3 times in a year. So the demand event rate is 0.25/Month. Each demand ...
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100 views

Normal approximation on (what it looks like) a poisson

I am self-studying inferential statistics from Larson's introductory textbook named: "Introduction to Probability Theory and Statistical Inference" (1st edition John Wiley & Sons.). I came ...
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69 views

Poisson Process - Determining Rainfall Accumulation

Disclaimer - I'm not a statistician. Most of my knowledge on the subject is self-taught. I'm trying to grasp how continuous random variables can be used in conjunction with a discrete Poisson process....
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42 views

Poisson Process: Probability of 2 arrivals by time 1 given that there are 6 arrivals by time 4?

Let $(N_t)_{t \ge 0}$ be a Poisson process with parameter $\lambda = 1.5$. Find the following: (c) $P(N_1 = 2 | N_4 = 6)$ I know how to do it if the times are reversed (it simply uses the ...
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Poisson Process: In a small parliamentary election, votes are counted according to a Poisson process at the rate of 60 votes per minute

In a small parliamentary election, votes are counted according to a Poisson process at the rate of 60 votes per minute. There are six political parties, whose popularity among the electorate is shown ...
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133 views

Poisson Process: Computers in a lab fail, on average, twice a day, according to a Poisson process

Computers in a lab fail, on average, twice a day, according to a Poisson process. Last week, 10 computers failed. Find the expected time of the last failure, and give an approximate time of day when ...
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1answer
45 views

Definition of (contiuous-time) Markov chain transition rate

Suppose that the rate at which a Markov chain leaves state i at some time t is $\lambda_{i}$. I.e., What is the rate at which $X_{t}$ leaves state i. Then, $\lambda_{i} = \sum_{j\neq i}q\left ( i,j \...
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426 views

PGF of a Poisson Process

I'm trying to find the probability generating function of a general Poisson Process and am a little stuck. The PGF is defined as $E(s^{N_t})$, and I know that the density function of $S_n$ is:$$...
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1answer
56 views

Limiting fraction of number of arrival while another event is occurring

A light bulb burns for an amount of time having distribution F with mean μF then burns out. A janitor comes at times of a rate Poisson process to check the bulb and will replace the bulb if it is ...
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1answer
330 views

What probability distribution would be suitable for modelling scores in a basketball match?

In football (not American football but what Americans call soccer), it is pretty clear: we may consider the difference of two independent Poisson variables. In basketball, in theory, we could increase ...
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Probability that epoch of kth arrival in first poisson process lesser than that of jth arrival in second poisson process

Let $S_{1k}$ and $S_{2j}$ denote the epoch of the $k^{th}$ and $j^{th}$ arrival in the first and second poisson counting process, respectively. The first and second poisson counting process is the ...
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136 views

Grid based piecewise-stationary Poisson process test

I'm trying to fit a set of data to a variety of Poisson-based models, and have hit a stumbling block when trying to fit a piecewise-stationary Poisson process. What I mean by this is a Poisson process ...
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1answer
461 views

Confidence interval for mean of Poisson distribution witih observations of different length

I am struggling with calculating a confidence interval for my data using a Poisson distribution. The context is manufacturing process quality control, and I think that a Poisson distribution is the ...
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2answers
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idea behind Poisson process property

A property of Poisson process says this: $N\left ( t \right )$ has independent increments: if $t_{0}<\cdot \cdot \cdot <t_{n}$ then $N\left ( t_{1} \right )-N\left ( t_{0} \right ),\...
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How to model continuous time discrete event rate

I have data consisting of irregular event times and at each event there is either a binary positive or negative result (with approximate probability of a negative result ~5%). The aim is to assess ...
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standard error & standard deviation (Poisson counts)

I observe 25 events over a period of 50 seconds. I assume that the events are generated by an underlying Poisson process. What is the standard deviation of the count? What is the standard error of ...
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1answer
61 views

Determining if an event is part of a poisson process

Trying to understand how to find an outlier in a Poisson process. Using example below to help me understand. A machine creates widgets at a rate of 10 per hour ($\lambda =10$) under normal ...
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544 views

Modeling the number of corners in soccer

For a project, I am looking for idea to model for the distribution of corners in football matches. I know that the number of goals can be model by a Poisson distribution, but for the number of corners,...
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179 views

simulation of non-homogeneous poisson processes

When simulating non-homogeneous (time-dependent hazard $\lambda(t)$) Poisson processes, we can use thinning [1] to generate jump times $t \in [0,t^0]$: initialize $t = 0$ generate $u_1 \stackrel{d}{\...
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Complete randomness imply Poisson process

In this paper, the authors claim that Theorem 1. A random point process $\Pi$ on a regular measure space is a Poisson process if and only if $N_\Pi$ defined by $N_\Pi(A) = \#\{\Pi \cap A\}$ is a ...
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The probability of disc that fully overlaps discs in a 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 disc of fixed radius $r$ is placed. Similar to ...
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Maximal aggregate loss in Risk Process

I am studying the Crámer Lundberg Risk Process. This process is defined as follows: $$Y_t=r_0+ct-Z_t$$ Where $Z_t$ is a compound Poisson sum, it means that $Z_t=\displaystyle\sum_{i=0}^{N_t} X_i$, ...
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213 views

Example of Memorylessness of a Poisson process

Could someone please clarify the following example (it is taken from Introduction to Probability by Bertsekas): When you enter the bank, you find that all three tellers are busy serving other ...
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422 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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406 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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176 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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76 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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336 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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225 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
300 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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66 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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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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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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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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638 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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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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429 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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173 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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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 ...