# 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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### 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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### 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 . 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. ...
### 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 ...
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 ...