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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Poisson distribution for customers arriving

Customer arrive at a mean rate of 20/hour, assuming a Poisson process, what is the probability that the shopkeeper will have to wait more than 5 minutes for the next customer?
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Equality of a probability of a Poisson process

I'm completing a proof about constructing a Poisson Process, let $X_i \sim \exp(\lambda)$ for $i =1, \ldots n$ (all independent from each other). I've shown that their sum $S_n = \sum_{i=1}^n X_i$ is ...
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Palm formula for Poisson processes

I'm reading some lecture notes and the following gets stated without proof If $N \sim Poisson(\lambda)$ and $f$ and $G$ are functions of $x \in S$ and of $(x, N)$ respectively, then $$ \mathbb E \int ...
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Expecation of $S(t)=\sum_{i=1}^{N(t)}(X_i+T_i)^2$ for Poisson process $N(t)$

Let $$S(t)=\sum_{i=1}^{N(t)}(X_i+T_i)^2$$ where $N(t)$ is a homogenous Poisson process with rate $\lambda=2$, $X_i$ iid with density $f(x)=e^{-x},x\geq0$ and $T_i$ are the arrival times of $N(t)$. $...
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poisson process help applying equations.

Let $\{N (t), t ≥ 0\}$ be a Poisson process with rate $λ.$ Let $T_1 ={}$time of the first event, $T_n ={}$elapsed time between the $(n − 1)$-th and the $n$-th event, $S_n ={}$the time of the $n$-th ...
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Distribution of $X_{N(t)+1}$ in poisson process [closed]

Assume $\{N(t)\}_{t\geq 0}$ is a poisson process with parameter $\lambda$, $X_n$ is the $n^{th}$ interarrival time, $n \in \{1, 2, 3, ...\}$, which means $X_n$ is exponential distribution with ...
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Poisson arrival conditional probability

A meteorite shower is a poisson arrival with a rate of 16.6 per minute. Given that 7 meteorites were observed during the first minute, what is the expected value of the time passed until the 10'th ...
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Application of memoryless property of exponential distribution

Events, occurring according to a Poisson process with rate $\lambda$, are registered by a counter. However, each time an event is registered the counter becomes inoperative for the next b units of ...
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Time continuous discrete states Poisson processes, why is this piecewise constant?

So, I'm studying time continuous discrete states stochastic processes and specifically I started looking into my notes about Poisson's processes. In my notes (from my university class), we define it ...
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Conditional Probability in a fishy Poisson process

Suppose a machine is recording a particle emission process of a piece of radioactive material.Once particles are emitted by the radioactive material, the machine will record immediately.It is known ...
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Joint distribution and covariance of Poisson process and waiting time

Hi I am having a trouble solving for this problem where I have to find 1) Joint distribution of $W_{1}$, $W_{r}$ for $r\geq2$. 2) $\operatorname{Cov}(W_{1},W_{r})$ for $r\geq2$. [Notation ...
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Poisson Arrival Process and Uniform Distribution

I'm brushing up on some basic probability and have this question: If we have a Poisson arrival process with arrivals $A_{1}, A_{2}, \dots$, and we know that there is one and only one arrival in a ...
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differential equation with a Poisson-process driven variable

I am trying to solve the following differential equation: $$dP_{t}=\beta K_{t}dt-\alpha P_{t}dt$$ where the variable $K_{t}$ is driven by a Poisson process $q_{t}$ as follows: $$dK_{t}=gK_{t}dt-(1-b)...
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Find the distribution of numbers of arrivals of the Poisson process $N(t)$ in time interval $[t, t+\tau)$, $\tau \sim Exp(a)$.

Poisson process has rate $\lambda$ and $\tau \perp \!\!\! \perp N(t)$. To find distribution i've started with $P(N(t+\tau)-N(t)=k) = P(N(\tau) = k)$. I know that $N(t) \sim Poiss(\lambda t)$, but i ...
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Help with Poisson Stochastic Process [closed]

Cars pass along the road in accordance with the Poisson process of intensity $\lambda$ . A pedestrian crosses the road at time $W$ as soon as he sees that there will be no cars during time $T$ (...
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Conditional expectation of poisson procces problem

Let $N_{t}^{i}$ - be three independent Poisson processes of intensity $1$. $\tau$ = $\inf\{t: \,N_{t}^{3} = 1\}$, $X^{i}$ = $N_{\tau}^{i}$ (means that $X^{i}$ - the values of the first two processes ...
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Exercise on Poisson processes

I've been trying to solve an exercise related to Poisson. This is the exercise: This is what i did (i am not sure about my answers from points 4 to 10) and honestly i do not understand point 1,2 ...
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How to prove that Markov chain with specific transition probabilities has independent increments?

I have Markov chain $N=\{N(t) \mid t\geq 0 \}$ with the state space $\{0,1,2,\dots\}$. I know that it is homogeneous and transition probabilities are: $$ p_{ij}(s,t)=P(N(t)=j\mid N(s)=i) = p_{i,j}(t-...
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Mathematical expectation of the number of points for Poisson stohastic process

I need to find mathematical expectation of the number of points of Poisson process with parameter $\lambda > 0$ such that: these points $\in [1,2]$ and there are no points of Poisson process in ...
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Conditional Probability of Sum of Poisson Point Processes

I'm having trouble determining the conditional probability of the sum of two independent Poisson Point Processes, $X, Y$, with parameters $\lambda,\mu$ respectively. If $X+Y=W$, I would like to find: ...
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Integral Ito (stohastic integral) of Poisson process

I need to find $D\int_{0}^{t} N_{s}dW_{s}$, where $N_{s}$ is Poisson process and $N_{s}$ and $W_{s}$ are independent. How can i do it?
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Stochastic Process problem. Poisson procces

Couldn't solve one exercise. Exercise is as follows: The Bank employs $10$ operators. The service time for each is exponentially distributed, with the average $i$-th operator serving $i$ clients per ...
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Poisson process, find the probability

I'm trying to prepare for my Stohastic Processes exam, and i couldn't solve one of exercises. The exercise is as follows: Costumers arrive to the store according to the Poisson process $N_t$ with ...
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Justifying Probability Assumptions in Derivation of the Poisson Point Process

I am finding it difficult to understand the justifications for an assumption made in some derivations of the Poisson point process. I understand, for a formally defined Poisson point process with rate ...
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One Poisson process arrives before the other: explanation of method

Let $𝑋_𝑡$ and $Y_t$ be two independent Poisson Process with rate parameters $\lambda_1$ and $\lambda_2$ respectively, measuring the number of customers arriving in stores 1 and 2. What is the ...
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Customers arrive at a service center according to a Poisson process with a mean interarrival time of 15 minutes.

Customers arrive at a service center according to a Poisson process with a mean inter-arrival time of 15 minutes. What is the probability that no arrivals occur in the first half hour? What is the ...
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Calculate $\mathbb{P}(P_{t+s} = n | P_s = k)$

Let $(P_t)_{t \geq 0}$ be a Poisson process with $\lambda >0$. Thus, we know that for $t,s \geq 0$ $$P_0= 0$$ $$P_{t+s}-P_s \sim \text{Poisson}(\lambda t)$$ I want to calculate $\mathbb{P}(P_{t+s} ...
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Calculate conditional probability of Poisson process

Let $\{N(t), t \geq 0\}$ be the Poisson process with $\lambda=1$. We know $$N(a,b) = N(b)-N(a)$$ $$\mathbb{P}(N(a,b) = n) = \frac{(\lambda(b-a))^n}{n!}e^{-\lambda (b-a)}$$ Now I want to compute $\...
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Sum of product of binomials

While working on a combinatorics problem, I found that this result had to be true: $$\sum_{i=0}^n\frac{(a-b)^i(b-c)^{n-i}}{i!(n-i)!}=\frac{(a-c)^n}{n!}$$ for $a\geq b\geq c$, with $a,b,c\in\Bbb N$. ...
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Stochastic Calculus for Jump Processes: Squared Compound Poisson Compensated Stochastic Integral

I'm reading these notes on stochastic calculus for jump processes which are great. On page 670 of these notes, the author derives the expected value of the squared compound Poisson compensated ...
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Can someone please explain me what a marked Poisson process is, and give an example of an application for this process.

Can someone please explain what a marked Poisson process is, and give an example of an application for which a marked Poisson process might be a useful model. I've been looking for explanations on the ...
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Inverse of Poisson process.

Let $N(T)\sim \text{Pois}(\lambda T)$. My question is: $$\lim_{T\to\infty}E\Big[\frac{1}{N(T)}\Big]= \; ?$$ Is it zero or $\infty$?
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Subinterval counting distribution of a Poisson counting process

Suppose $N(\omega,t)$ is a [homogeneous Poisson counting process][1] with a constant parameter $\lambda,\,\forall\omega \in\Omega$ where $\Omega$ is the sample space. Given positive real numbers $T$ ...
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Buses arrive in Poisson rate at a bus stop with 0.6 probability it is the one you want

Buses arrive at 116th and Broadway at the times of a Poisson arrival process with intensity λ arrivals per hour. These may either be M104 buses or M6 buses; the chance that a bus is an M104 is.6, ...
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Laplace functional for Poisson Process: $E[e^{-\sum_{n=1}^{\infty}f(W_n)}]= e^{-\lambda\int_0^{\infty}(1-e^{-f(t)})dt}$

Let $W_n$ be the $n$ waiting time of a Poisson process. Proof that $$E[e^{-\sum_{n=1}^{\infty}f(W_n)}]= e^{-\lambda\int_0^{\infty}(1-e^{-f(t)})dt}$$ for a measurable postive function $f$ I really don´...
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Proof of non-homogeneous Poisson thinning

Currently, I am trying to prove the $\textbf{Theorem 3.7.5} $ in Durrett Probability theory and example. The Theorem is $\textbf{Theorem 3.7.5}$. Suppose that in a Poisson process with rate $\...
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Probability tricky exercise (Poisson)

I've been trying to solve this exercise about Poisson but it seems my reasoning is not correct. It goes like this: Failures in a system are modeled according to a Poisson process with a rate of $0....
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Problem of Composite Poisson Process

Accidental damage $Z$ in the classical risk model is distributed according to the law:$$F_z(x)=\begin{cases}0,&\text{if }x\le 100,\\\dfrac{x-100}{900},&\text{if }100<x\le1000,\\1,&\text{...
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Orbital motorway example in Poisson processes by Kingman

In Kingman's Poisson processes chapter 5, there is a modelling example of an orbital motorway. Basically, $\Pi'$ is a Poisson process on $[0,2\pi)\times (0, \infty)$ (where $[0,2\pi)$ stands for the ...
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Meteor hitting Poisson process question [closed]

Kate is monitoring her Traffic. She estimates that it is hit by about one car per week. You may assume that the times at which car hit the another car are described by a Poisson process.
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How do I generalize a certain Markov model?

This question is a further attempt to generalize a certain Markov model of limit & market orders arriving in a financial exchange as first proposed in [1]. See Solving another non-trivial ...
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Itô-Tanaka formula for a cadlag semimartingale (with a jump process)

Let $T<\infty$ and $Y$ be some cadlag semimartingale such that : $$ dY_t = r_tdt + \sigma_t dW_t + \beta_tdN_t,\ t\in [0,T] $$ Where $W$ is a $1$-dimensional Brownian Motion, $N$ be a jump ...
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Why does the infinitesimal of a poisson process behave as it does in the Ito multiplication table

In more informal derivations of Ito's formula for jump processes, the multiplication table for $dt, dW_t$ and $dN_t$give that $dN_tdN_t=dN_t$. Why is this? I have tried deriving this from the ...
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Exercise 2.4.1 norris

I am working though the book of J. Norris, "Markov Chains" as self-study and have difficulty with the second part of ex. 2.4.1. (Complete exercise) State the transition probability definition of a ...
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Poisson process exercise- two intervals with OR

I have trouble with the following exercize, expecially with the use of the “OR” conjunction between the intervals (perhaps suggesting intersection?) Let {N(t),t∈[0,∞)}be a Poisson process with rate λ....
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The number of customers in a store at certain time?

The question: Customers arrive in a coffee shop one-by-one according to a Poisson process with intensity $\lambda = 6$ per hour. They stay at the coffee shop during a random duration $V$ which is ...
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Sum of Poisson process and uniform distribution

I'm studying for my final and have the following question. I'm looking for clarification of my first answer and how to approach part b) and c). Men and women arrive at a store according to ...
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Finding Conditional Probability of Poisson Process, Markov Chains

Consider a Poisson process {X(t), t ≥ 0} with constant rate λ > 0. Let X(a, b) denote the number of events in the time interval a < t ≤ b, and 0 < t1 < t2 < t3 are three given time points. ...
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Find $\text{Cov}(N_{1}(t),N_{2}(t))$

Let $\{M_{i}(t), t \geq 0\}$, $i=1,2,3$ be independent Poisson processes with respect rates $\lambda_{i}$, $i=1,2,3$ and set $$N_{1}(t)=M_{1}(t)+M_{2}(t), \quad N_{2}(t)=M_{2}(t)+M_{3}(t)$$ The ...
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Need help with a Homogeneous Poisson Process Question

Customers arrive at a shop following a homogeneous Poisson process $N(t), t ≥ 0$, rate $\lambda$. Each customer spends some amount of time, $t_i$ in the shop, with mean $E[t_i] = \mu_t$. If there is a ...

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