Questions tagged [poisson-process]

Questions relating to the Poisson point process, a description of points uniformly and independently distributed at random over some space such as the real line. The number of points within some finite region of that space follows a Poisson distribution.

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Taylor Expansion of a function with random variables

Setting Let $X_t \in \mathbb{Z}^{0+}$ be a random variable at time $t>0$. We have two counting processes $N(t)$ and $M(t)$ with: \begin{align*} & N(0) = M(0) = 0, \\ & dN_t \sim \text{...
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$M_t = \frac{1}{\sqrt{T_1}} \mathbb{1} (T_1 \leq t) - 2 \sqrt{T_1 \wedge t}, t\geq 0$ is a martingale

I try to solve an old exam question, but I find it difficult. Maybe someone can suggest a hint. Let $\{ T_i | i\in \mathbb{N} \}\subseteq \mathbb{R} _{\geq 0}$ be a homogeneous Poission point process ...
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Probability of forming an n-gon [closed]

We have more than 3 lengths. Each length is exponentially distributed with parameter $\lambda$ and each is independent and identically distributed. I want to calculate the probability of constructing ...
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Conditional expected value of Poisson processes

Find Find $ E \{ \frac{ \xi(5) }{ \xi(7)+1 } \ | \xi(7) \} $ where $ \xi(t) $ is standard Poisson process I can easily solve similar problems, where a random process is represented by an ordinary ...
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A question on a compound Poisson process: $P(|\xi(t)|\leqslant0.3)\underset{t\rightarrow\infty}{\rightarrow}0$?

Let $\xi(t)$ be a compound Poisson point process such that the number of summands is a Poisson point process with parameter $\lambda$ and the summands are random variables: $P(\xi_k=\pm1)=0.5.$ Is it ...
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Confusion on when to use CDF and Poisson process

I'm going through the MIT OCW probability course (6.041sc), but I'm having trouble on when to use CDF and the Poisson process. Here's the problem (Recitation 15, problem 1). Problem Statement: ...
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Can't understand the proof of the Time-Rescaling theorem.

I was reading the following paper: The time-rescaling theorem and its application to neural spike train data analysis and I have some difficulties understanding the proof of the time-rescaling-theorem....
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If $X$ is a Poisson process, what is $\mathbb{P}(X(B_{a}(x)) = 0 \text{ for some }x \text{ with } |x|=b)$ for $0<a<b$?

Let X be a Poisson process with intensity $1$ on $\mathbb{R}^2$. For $0<a<b$, what is the value of $$\mathbb{P}(X(B_{a}(x)) = 0 \text{ for some }x \text{ with } |x|=b)$$ where $|\cdot|$ is the ...
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Why is a completely random measure mixing?

Let $S_x$ denote an operator on $\mathcal{M}_{\chi}^{\#}$ by $S_x\xi(\cdot)=\xi(\cdot+x)$, where $\xi$ is a random measure. Here, $\mathcal{M}_{\chi}^{\#}$ refers to the space of boundedly finite ...
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Poisson processes of two airline companies. (two independent Poisson processes)

Easyjet and KLM planes request landing permission at Heathrow airport according to independent Poisson processes with intensities $\lambda$ and $\mu$ per hour, respectively for Easyjet and ...
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Correlation of Markov process

I have a problem where cars are entering an area according to a homogeneous Poisson process, with a rate of $\lambda = 9$ cars per hour. 20% of the cars entering the area are of type 1, and 80% of the ...
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A customer queue where service time is exponentially distributed and arrival governed by a Poisson process.

I have just started Poisson processes in my course on stochastic processes. We have just covered random sums and now I got the following tutorial/ exercise class question, but I don't quite see what ...
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Poisson random measure and $\alpha-$stable processes

Let $N$ be a Poisson random measure in $(0,\infty)^2$ with intensity $\eta$ given by $$\eta(ds, dx) = \mathbb{I}_{\{x>0\}} \dfrac{C}{x^{\alpha + 1}} ds dx.$$ Find the values $\alpha$ for which $\...
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Stirzaker “Elementary Probability” shower problem

Your telephone is called at the instants of a Poisson process with parameter λ. Each day you take a shower of duration Y starting at time X , where X and Y are jointly distributed in hours(and not ...
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Hitting time of Poisson process is finite

let us consider Poisson process $N_t$ with $\lambda$ parameter and a stopping time $T=\inf\{t\ge 0;\,N_t=a\}$, where $a\in\mathbb{N}$. I would like to show that $ET=\frac{a}{\lambda}$, so I want to ...
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probability density of a nonhomogeneous poisson process

I am beginning to study Poisson processes and have come across this question involving a function $\lambda(u) = u + 1$ so that it is a non-homogeneous Poisson process. Let $\tau$ be the time between ...
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phenomenon of bulk of arrivals in poisson process

"the phenomenon of bulk arrivals can be handled by the simple extension of associating a positive integer RV to each arrival" (stochastic processes by Robert g. Gallager) I search a lot for ...
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On the sum of two Poisson- processes.

Suppose $(Z_t)_{t\in\mathbb{R}}$ is the sum of two independent Poisson processes $(X_t)_{t\in\mathbb{R}},(Y_t)_{t\in\mathbb{R}}$ with rates $\mu,\lambda$ respectively. I know I have to show the ...
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How to derive Negative Binomial process from Poisson process?

I am trying to understand NB process, and how it can be derived from Poisson process. Zhou & Carin states that: "By placing a gamma prior with shape $r$ and scale $\frac{p}{1−p}$ on $λ$ as $m ...
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Generation of a Poisson process

I am having trouble understanding the Poisson process generated as the following (in MATLAB): Choose the number of points, i.e., $N$ and a parameter $\mu$; Compute $y = rand(N,1)$, i.e., $N$ random ...
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counting processes, stationary and independent increments

I know that Poisson process has stationary and independent increments. Are there any other known discrete processes with these two properties?
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Expectation of Integral of Stationnary Process

Let $(X(t))_{t \geq 0}$ be a Poisson process, $Z$ a Bernoulli random variable, independent of $X(t)$. We define $$Y(t)=Z(-1)^{X(t)}$$ It is clear that $Y(t)$ is sationary. Now we define $$S(t)=\int_{0}...
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Given the inter-arrival distribution of busses, what is the distribution of time until your bus arrives?

This is one of those "share your knowledge" posts. I have an answer (which I'm going to post shortly), but would greatly appreciate if people could point out any issues with it or provide ...
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Is natural filtration of a left-continuous modification of a Poisson process is right-continuous?

I know that a natural filtration of a Poisson process is right-continuous. But does the same hold for a left-continuous modification of that Poisson process (denote $N'$). My guess, it does not hold. ...
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Poisson Process timed on other arrivals

I've been struggling on this question for a while. Particularly, part b. My thought process was that it uses 2C1 competing exponentials to end up with that result, although it does not make intuitive ...
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Derive an expression for the probability that there are k tasks that are half-done at the instant when exactly one task becomes completely done.

There are n tasks that are given to n processors. Each task has two phases, and the time for each phase is given by an exponentially distributed random variable with parameter 1. The times for all ...
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Prove that $S_N = \sum_{i=1}^{N} X_{i}$ is exponentially distributed with parameter $p$ [duplicate]

Let $X_{1}, X_{2}, \ldots$ be exponential random variables with parameter $1 .$ Let $N$ be a geometric random variable with parameter $p$. Prove that $S_N = \sum_{i=1}^{N} X_{i}$ is exponentially ...
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Determining $\lambda$ and $\mu$ for a queuing system

I have just started studying the topic of queuing systems and I am having trouble with grasping the intuition that I need to start solving questions. Consider a M/M/1 queue where arrivals occur as a ...
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Deriving Poisson arrival process from operation times

I have a problem from my Stochastic Processes class I have no idea how to solve. We have been studying Poisson processes including models of queues, but this question involves a small amount of sample ...
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Not sure about an exercise of probability (POISSON)

I was given this exercise: I did this: I know that: Fabric rolls of 10 meters² and according to POISSON 2 defects are expected for every 20 meters². Having 2 defects every 20 square meters means ...
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Poisson process vs. Poisson Random Measure

Let $(W_i)_{i \in \mathbb{N}}$ be iid Exp($\lambda$) and $T_n := \sum_{i=1}^n W_i$. Then it is well-known that $$ N(t) := \sum_{n \in \mathbb{N}} 1_{T_n \leq t} = \sum_{n \in \mathbb{N}} \varepsilon_{...
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Intuition Behind Independence of Thinned Poisson Processes

Let $(N_t)_{t \geq 0}$ be a rate-$\lambda$ Poisson process, and let $(X_i)_{i \geq 0}$ be IID $\text{Bernoulli}(1/2)$. Denote by $(N_t^i)_{t \geq 0, i \in \{0, 1\}}$ the thinned Poission processes ...
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Need help validating a proof that for any point process with MTBF $t$, the events in an interval sized $u$ will be $\frac{u}{t}$

I started a bounty on this question here: General point process - expected number of arrivals within an interval. The premise is that we have a point process where the time between successive events ...
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Statistics - Binomial and Poisson Distribution Problem

I am given: I get 11 text messages per hour according to a Poisson process. The probability that a given text message is from my mother is $0.62$. I then have to find the probability that I receive ...
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Change of jump sizes in Poisson processes

After applying Ito's lemma, I arrived to the following stochastic differential equation: $$ dg(Y_t)=\left[ g(Y_t+t)-g(Y_t)\right]dN_t + g^\prime (Y_t)\beta^t \lambda dt$$ where $g$ is continuous, $N_t$...
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Finite intensity of Lévy measure implies compound Poisson process

Suppose $X$ is a Lévy process with triplet $(b,\sigma^2,\nu)$ and finite intensity, so $\nu(\mathbb R)<\infty$. Why does it follow immediately that the jump part of $X$ can be described by a ...
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Simultaneous Poisson Processes of Cabs and Passengers

Consider a airport exit with people leaving with a poisson process rate of 1 per minute. From the exit they can catch a cab, which arrives with a poisson process of 2 per minute. A person will wait no ...
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Calculating probability for exponentials

Let $T_1, T_2$ be exponentials with rate $\lambda_1, lambda_2$. We want to find $P\left(T_{1}<T_{2}+T\right)$. I did: $P\left(T_{1}<T_{2}+T\right) = \int_0^\infty P(T_1 < t + T) f_{T_2}(t) dt ...
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The understanding of Poisson process

I am learning Poisson Process right now and confused about some concepts about it. First, we know a stochastic process has two parameters, the event $\omega$ and the time $t$. For a given $t$, the ...
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Conditional expectation probability mass function

In a probability example, there are these random variables: $X_i, N_t,$ and $Z_t=E(\sum\limits_{i=1}^{N_t} X_i | N_t)$, where $S(N_t)=\mathbb{N}_0$ and $X_i$ is a continuous random variable. I have ...
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General point process - expected number of arrivals within an interval

We have a point process where the interarrival time is described by a random variable, $T$. Further, $E(T)=t$ and $T=t+\epsilon$, where $E(\epsilon)=0$ and $\epsilon \in (-t, \infty)$. The $\epsilon$ ...
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Poisson process. A problem about customer arrival at different time intervals. Wackerly 3.125

I was solving exercise 3.125 of Wackerly's Probability book and i did not understand the solution given in the solutions manual. The problem says: Customers arrive at a shop following a poisson ...
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PGF and the Kolmogorov backward equation

$(N_t,t\ge 0)$ is a poisson process with rate $\lambda$ Using the Kolmogorov backward equation, find the PGF: $$G_i(z,t)=\Bbb E[z^{N_t}|N_o=i]$$ The answer is $G_i(z,t)=e^{\lambda t(z-1)}$ but I am ...
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The sum of $n$ independent Poisson random variables is a Poisson random variable itself: explanation not proof.

I understand how to prove that the sum of $n$ independent Poisson random variables is a poisson random variable mathematically. However, I don't understand this in terms of the concept of Poisson ...
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Poisson distribution for arriving vehicles

I'm interested to model the number of Electric Vehicles (EVs) which arrive to a charging station during one day and their Time-of-Arrivals (ToA). I read that the number of EVs arriving at a charging ...
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The expectation of sum of squares of the arrival times in a Poisson process

Question: Let $\{N(t)\}$ be rate $\lambda$ Poisson process, with arrival times $\{S_n,n=0,1,...\}$. Evaluate the expected sum of squares of the arrival times occuring before $t$, $$E(t)=\mathbb{E}\...
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Poisson Process Service Queue

Consider a single-server queue with Poisson rate λ arrivals and Exponential rate μ>λ service times, but make the following modification: Fix some α∈(λ/μ,1).When a service time is complete, with ...
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Thinning in Poisson Process

please help in providing a proof like explanation. I am confused and do no know where to start. Let $(X_t)$ and $(Y_t)$ be independent Poisson processes with rates $\lambda$ and $\mu$. Using thinning/...
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Covariance between exponential and gamma distribution

Define $W_{r}=\min\{t:N_{t}\geq r\}(r=1,2,...)$ as the waiting time until the $r$th arrival in a Poisson process $\{N_{t}:t \geq 0\}$ with rate $\lambda$. The question is to find $Cov(W_{1},W_{r})(r\...
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Poisson Point Process Closest Point

Suppose we have a Poisson point process, with intensity $\lambda$. I want to prove that the expected distance from the origin and the closest Poisson point is $\frac{1}{2\sqrt\lambda}$ when in two ...

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