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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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Simulate an arrival time - Cox Process?

If I simulate arrival times using a Poisson process where the input to the Poisson process is also a Poisson process. Input arrival rate = L , and then U1 and U2 are random draws from a uniform ...
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20 views

Point process - intensity function vs probability density function

Suppose we have a point process in $\mathbb{R}$ with intensity $\lambda(x)$. Then, for a given compact set ${ S}$ we have $$\Lambda({ S})=\int_{\rm S} \lambda(x) \, dx,$$ where $\Lambda({ S})$ is ...
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43 views

Probability of an event with multiple conditions

Could you inform me please, how can I calculate conditioned probability of several events? I have 3 events A, B, C; I know P(B|C) and I want calculate P(A|B,C). Is it possible? In the special case ...
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6 views

fix the number of class in random poisson simulation

I want to simulate a poisson distribution in R with rpois and in my generated vector I want to have a fixed number of level example ...
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15 views

LRS for poisson process

How to use likelihood ratio statistics test for Poisson process with different intensity function model? Model 1: $ \lambda(t)=\lambda$ Model 2: $ \lambda(t)=\exp( \beta_1 + \beta_2 t)$ I think the ...
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26 views

Does exponential waiting time for an event imply that the event is Poisson-process?

Say I have a process, $\{N_t : t \ge 0\}$, which denotes the number of the event that occurred until the time $t$. And let me define $W = \min \{t : N_t = 1\}$ which is denotes the time until the ...
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597 views

Nonhomogeneous poisson process simulation

I've been looking at ways to generate a Nonhomogeneous Poisson Process (NHPP) including the nonlinear time transformation (using a rate-1 process and inverting the cumulative rate function). I've also ...
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1answer
34 views

Conditional expectation of Poisson process given number of events

Let $\{N(t), t\geq 0\}$ be a Poisson process with rate $\lambda$, $S_n$ the instant of the $n$-th arrival and $T_n$ the $n$-th interarrival time, that is, $T_n = S_n - S_{n-1}$, $n \geq 1$. Now ...
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42 views

Poisson process and queuing system

In a queuing system, the inter-arrival times are known to be exponentially distributed. My textbook states: "It can be shown that, if the underlying distribution of inter-arrival times { T1, T2, ..., ...
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1answer
237 views

Non homogenous Poisson process with simple rates

I am trying to stimulate number of claims in the next 12 months using a non-homogeneous poisson process. The rates are: ...
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2answers
416 views

Trajectory of homogeneous poisson process

I am trying to simulate number of claims in next 12 months using a homogeneous poisson process following the R codes: ...
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34 views

Problems on Poisson process

Consider a homogeneous Poisson process with inter-arrival times $T_i$, which follows the exponential distribution with rate $\lambda$. Let $N(t)$ denote the number of arrivals by time $t$. Suppose I ...
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Error prediction for poisson process

A Poisson process has rate of N per unit length per unit time. The events occur uniformly on the x-axis but I always have measurement error of greater than 1/N. I was wondering is it possible that I ...
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1answer
91 views

Gamma Conjugate Prior & Poisson Process

I am analyzing daily data transaction data. I am assuming that The number of transactions in every day of length t has the Poisson distribution with mean λt The number of transactions in evert ...
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1answer
24 views

Poisson process (find mean) - solution check

Mary receives 22 messages in 5 hours according to a Poisson Process. What is the mean number of messages Mary receives in an hour? Seems like a simple question, so just to confirm: Mean = 22/5 Isn'...
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40 views

Binomial/Poisson Problem - Pizza Orders

Suppose that the number of orders per hour at a pizza shop follows a Poisson process with rate 5 per 30 minutes. Suppose that the pizza orders are large with probability 2/3, small with probability 1/...
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25 views

Poisson/binomal problem - car speeding [duplicate]

A certain police officer stops cars for speeding. The number of red sports cars she stops in one hour is a Poisson process with rate 4, while the number of other cars she stops is a Poisson process ...
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49 views

Conditional probability with poisson (can this be solved by binomial?)

Pizza orders arrive according to a Poisson process of rate $20$ per hour. Orders are independently for a vegetarian pizza with probability $\frac14$ , and for a meat pizza with probability $\frac34$. ...
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25 views

Probability of stale cache hits w/ exponentially distributed read and write inter-arrival times

I am trying a create a model for a very simple system which consists of a server and 2 clients: R and W. The server stores a piece of data D and hands out leases for caching D on the clients. The ...
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1answer
19 views

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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38 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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1answer
90 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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10 views

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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24 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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1answer
28 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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11 views

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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141 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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112 views

Help me understand poisson.test?

I want to understand the poisson.test() function: ...
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51 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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67 views

Waiting time paradox - rigorous math description

I was reading about the waiting time paradox. I can understand that from the memoryless property of the exponential distribution, if the events occur with rate $\lambda$ then the average waiting time ...
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13 views

How to choose time bin width for Poisson Process?

Newbie here - just started learning more about poisson distribution. I have a time series of occurrence of events, and I thought this might fit a Poisson process. Before I do hypothesis testing, I ...
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18 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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1answer
63 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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1answer
56 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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26 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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1answer
62 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
42 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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182 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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30 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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83 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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103 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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334 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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10 views

Error on a ratio of two measured values each with just a poisson noise estimate

I am trying to estimate the error on some discrete data. I have a total of N counts of which x have the desired characteristic. So x/N is the fraction of the sample I want to plot. I am assuming ...
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43 views

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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384 views

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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206 views

Difference between standard deviation and standard error (for low count rates)

Similar questions about sample means and counts, but neither directly about count rates. I get $\lambda=4$ counts over a $T=25$ day interval. If I assume that the counts are yielded by an underlying ...