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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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Simulating a (discretized) Cox process via binomial sampling

Let X be a Cox process (doubly-stochastic Poisson process) with fixed intensity(rate) $\lambda=50$ , and choose some small time interval $dt=0.01$ . Is the proper way to simulate this, by letting Y ...
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Inhomogeneous K-function to indicate need for spatial dependence/interaction term in Poisson point process model

I am mapping and modelling a disease of sheep. I have approx 4200 point locations in my dataset, each of which represents the centroid of a given sheep farm. I have created a K-function difference ...
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Proof for simulation of NHPP by thinning

Background: I'm trying to show equivalency between the density function for a non-homogenous exponential process (NHEP?), (i.e. the arrival times of events generated by a non-homogenous Poisson ...
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two independent Poisson Arrivals

I have two types of customers (type 1 and type 2) enter a shop. Their arrival processes are independent and follow Poisson process with the arrival rates of $\lambda_1$ and $\lambda_2.$ Consider two ...
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Expected time to wait for no events to occur within a sliding window assuming Poissson process

I wish to model the following: I am maintaining a sliding window (history) of 10 samples of the output of a signal detector. I model the probability of a detection failure (i.e absence of signal) as ...
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Poisson Process Simple Question

The number of customers that arrive at a cashpoint in an hour is distributed poisson($\lambda$). Suppose that each arriving customer makes a draft. Let $Y_i$ denote the amount of money $i^{th}$ ...
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128 views

Probability Density function of Poisson distribution

This is an assignment I got for my course on Stochastic Processes: Let us consider a random variable X distributed as a Poisson P (λ) where λ ∼ [0.5, 1]. (a) Which are the unconditional ...
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Poisson and conditional probability

Admit that the number of participants who intend to enroll in a given training follows a Poisson distribution with a mean of $12.$ If there is not a minimum of five enrollments, training is not ...
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Compound risk poisson models

I was just working through this question. A compound Poisson risk model is used to model the total claims S experienced by an insurance company over one year, of the form: $S = X_1 + ... + X_n$ ...
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Intensity function in Poisson random effect model

I have a somewhat general question about intensity functions in Poisson random effect models. Consider the Poisson random effects model in which conditional on a random effect $u$, an individual ...
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Coverage of a Boolean model

The following problem I am working on involves the coverage of Boolean Model (BM). The grains in the BM are discs with fixed radius $r$. $A$ is the set of this BM in space $ \mathbb R^2 $ and $K$ ...
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Correcting data using poisson-regression

I'm new to stats and I was wondering if anyone had any good resources that could explain to me: How one can correct their data (false-positives) using Poisson-regression. I've been looking for some ...
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How to model arrival times in discrete event simulation where arrivals vary with time of day and day of week

I'm looking for suggestions on how to model inter-arrival times in a discrete event simulation where the arrival rate is highly dependent on the day of week and the hour of day. For example: A ...
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The probability distribution of waiting time until two exponentially distributed events with different parameters both occur

I am working on a problem related to the waiting time until a parking garage is empty. We are given that the cars independently spend an exponential distributed time in the parking garage, with ...
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Partial imputation of missing dates

I'm working with dataframes (one for each of 185 locations) that shows sums of occurrences for each calendar date. There are no 0 values for occurrences in the entire dataset. There are several ...
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Computing probabilities of comparisons of exponential random variables

I am working on a question for class where there are two patients waiting on kidneys. The arrival of transplant kidneys is a Poisson process with rate λ. A will die after an exponential time with rate ...
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question regarding a stochastic process

Say let $Y_1, Y_2,...$ be a family of i.i.d random variables with a common $\mu$ and common variance $\sigma^2$. Let $N_t$ be a Poisson process with rate $\lambda >0 $. Assume that $N_t$ is ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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236 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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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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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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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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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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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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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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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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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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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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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 ...