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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Extension of Poisson Parameter for Different Temporal Interval

Suppose $Y$ follows a Poisson distribution with parameter $ \lambda $ that explains a temporal Poisson process over an interval of 30 seconds. Now, it stands to reason that for an interval of 60 ...
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Test and CI for unstable Poisson process

I have empirical data on a process that I assume is Poisson with a given mean, say $\mu$ (unknown). The data is of the form $(x_i, 1\leq i\leq n)$ for $n$ consecutive time periods. I am concerned ...
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Upper confidence bound for Poisson process rate parameter

I am interested in computing an upper confidence bound for the rate parameter, $\lambda$, in a Poisson process. Specifically, I have a set of observations $$ X_\text{obs} = \{(n_1,t_1), \ldots, (n_N,...
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Alternating between two states {A, B} each with exp distributed durations. What's the probability of state=A at time t?

Say I have a light bulb that can be on (A) or off (B). It alternates between being state A or B. It will be in state A for a duration a ~ exp(α), and in state B for duration b ~ exp(β), (...
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Conditional distribution of arrival times in Poisson process

Suppose I know over a window $[0, T)$ that I have observed $n$ samples from a poisson process $N_t \sim p(n|\lambda t) = \frac{1}{n!}(\lambda t)^{n}\exp(-\lambda t)$. What is the conditional ...
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Poisson Process With Two Different Rates

I am looking to investigate a Poisson Process that has two separate rates; essentially, there is a season of games that each have their own points rate (per minute). Comprehensively, over the season, ...
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15 views

Overdispersion under different longitudinal constraints

I have panel data (police districts observed across many months) and I am modeling a count outcome. Fitting a Poisson model (simple pre/post comparison with 6 months of data) results in significantly ...
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17 views

How to compute the expected number of events in the following conditional renewal process?

I have a stochastic point process with event times $\{x_1, x_2, ...\} $ and I want to compute the expected number of events $n(T)$ over the interval $[0,T]$. The point process is generated as follows: ...
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How to show the inter-arrival time variance of a Cox process driven by a stationary Poisson process of constant intensity $\lambda$ is $3\lambda$

Ideas on how to show that the variance of a doubly-stochastic Poisson process(aka a Cox process) driven by a homogeneous(stationary) Poisson process of intensity $\lambda$ is $3\lambda$ ? I've come ...
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29 views

To detect which of the three processes more-aptly fits Poisson process

There happens to be a type of generic poll, which can be described as:- A participant can vote only once and for only 1 of 3 running candidates - Red, Blue and Green. A random voter starts the ...
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Solving an equation with logarithms of negative numbers

I'm modelling a bunch of non-homogeneous poisson processes. I'm impossing on them a linear-log functional form. you end up with two parameters, theta1 and theta0 for each estimated process. The ...
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Understanding the solution to a problem about a homogeneous Poisson process

This is probably easy, but right now I can't figure it out, so bear with me. The question is: Let $\{N(t),t\ge 0\}$ be a homogeneous Poisson process on $(0,\infty)$ with rate $\lambda$. Let $\{S_i, i=...
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Manually simulating Poisson Process in R

The following problem tells us to generate a Poisson process step by step from $\rho$ (inter-arrival time), and $\tau$ (arrival time). One of the theoretical results presented in the lectures ...
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Is this problem worked out correctly? [closed]

Calls are received at a company call center according to a Poisson process at the rate of five calls per minute. (a) Find the probability that no call occurs over a 30-second period. (b) Find the ...
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Expectation of arrival times

Let $(N_t)_t$ be a Poisson process with parameter λ = 2. By $τ_k$ denote the time of the k-th arrival (k = 1, 2, . . .), and by $ρ_k = τ_k −τ_{k−1}$ - the interarrival time between the (k−1)th and kth ...
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How to check if a data is poisson sampled?

I was reading one article which develops a theory for the Poisson sampled data. That is the data is collected over time-points $\{T_k, k>1\}$, which are jump-moments of a homogeneous Poisson ...
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52 views

What is the difference between time, arrival-time, and inter-arrival-time is poisson process?

Let $(N_t)_t$ be a Poisson process with parameter λ = 2. By $τ_k$ denote the time of the k-th arrival (k = 1, 2, . . .), and by $ρ_k = τ_k −τ_{k−1}$ - the interarrival time between the (k−1)th and kth ...
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How to properly truncate exponential distribution to represent random memory-less arrivals?

Typically, Poisson and exponential distributions are used to represent random memory-less arrival processes. It has come to my attention, however, that a more realistic distribution is a truncated (or ...
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25 views

Distribution of the y-coordinate of a 2D Poisson Process

Consider a Poisson process with rate 1 over $R_+ \times R$. I found this term in a scientific article, I assume it means : -The number of points in a subset $A$ of $R_+ \times R$ has the Poisson ...
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23 views

Generating Arrival Counts Given Time

I'm looking to generate arrival counts given a length of time for a Poisson process. This is similar to another question, however rather than generate the particular arrivals, I'd like to just get the ...
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Poisson process - measuring empirical changes

I was recently discussing Poisson processes in industrial settings with a colleague and he came up with a great question. Let's say that defective products on a manufacturing line tend to occur ...
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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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67 views

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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251 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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162 views

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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101 views

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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109 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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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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423 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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673 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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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'...