# 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.

682 questions
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### Probability of Ruin at the first claim

The number of claims $n \sim Po(\lambda)$, and let $X_n$ denotes the claim amounts of a claim which are all iid and they follow a $Exp(1)$ distribution. Assume the initial surplus is $U$, and the ...
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### Order non-homogeneous Poisson process rate functions by specific (end) time

The broader problem I am trying to solve is the following: Given $k$ lists of arrival times (all times less than or equal to some end time $T$) coming from $k$ non-homogeneous Poisson processes with ...
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### Poisson Process: Time until next arrival

Question: Suppose that busses arrive at a bus stop as a Poisson process with rate $\lambda$ starting from time $t=0$ (that is, the interarrival time between busses is exponentially distributed with ...
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### Compensator of a non-homogenous Poisson process

Given a non-homogenous Poisson process $T_n$ with intensity $\lambda(t)$ and the compensator $\int_0^t\lambda(t)ds$ How can I show that $\int_0^t\lambda(s)ds$ is predictable?
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### predictable projection of non-homogeneous poisson process

How to prove that the compensator of a non-homogeneous poisson process is the predictable projection of this non-homogeneous poisson process?
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### Poisson Process with Stationary Arrival Rate - Conditional Arrivals

I am having trouble with the following, We have a Poisson process that operates 24 hours per day with an arrival rate $\lambda = 3$ per hour. Suppose an observer arrives at 3pm. By 5pm, the observer ...
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### Construction of probability measure for Hawkes process

How to construct a probability measure for the Hawkes process? Like here
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### Find expected value of compound Poisson process

We have compound Poisson process with $\lambda = 3$ and jumps $D_{j}$ of size $1$ or $2$, where $P \{ D_{i} = 1 \} = 0.25$. Find mean value of this process in $t = 10$. Mean value of compound Poisson ...
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### expected value from some points in continuous homogeneous spatial Poisson point process

Let $n$ point are distributed as per a homogeneous spatial Poisson process of rate $λ$ in a square of side $2a$, and assume that $4$ fixed points are located at $(a/2,a/2)$, $(-a/2,a/2)$, $(a/2,-a/2)$ ...
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### Poisson point process representation

Let $\Pi: ( \Omega, \mathcal{F}, \mathbb{P} ) \rightarrow \mathbb{R}^d$ be a Poisson point process. We know that $\Pi_0=\{\left \| X \right \|, X\in \Pi\}$ is a Poisson point process on $\mathbb{R}_+$...
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### First event distribution of a non-homogenous Poisson Process when the event is not bound to occur

Let $N(t)$ a non-homogeneous Poisson Process with rate $r(t)$. If the rate sum up to infinity, i.e. $\int_0^\infty r(u)du=\infty$, the first event distribution, $f_{T_1}$ can be expressed as: \begin{...
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### Probability that $25$ calls are received in the first $5$ minutes.

Calls are received at a company according to a Poisson process at the rate of 5 calls per minute. Find the probability that $25$ calls are received in the first $5$ minutes and six of those calls ...
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### Show that the expected total present value of the bonds > purchased by time $t$ is $1000\lambda(1-e^{-rt})/r.$

Investors purchase $1000$ dollar bonds at the random times of a Poisson process with parameter $\lambda$. If the interest rate is $r$, then the present value of an investment purchased at time $t$ ...
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### Probability that at least one other vehicle arrives between third and fourth var arrival.

Starting at $6$ a.m, cars, buses and motorcycles arrive at a highway toll booth according to independent Poisson processes. Cars arrive about once every $5$ minutes, buses about once every $10$ ...
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### Let $(N_t)_{t\geq 0}$ be a Poisson process with parameter $\lambda=2$ find $\mathbb{E}[N_3N_4].$

Let $(N_t)_{t\geq 0}$ be a Poisson process with parameter $\lambda=2.$ Find $\mathbb{E}[N_3N_4].$ The solution here is \begin{align} \mathbb{E}[N_3N_4]&=\mathbb{E}[N_3(N_4-N_3+N_3)]\tag1\\ &...
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### Finding expected value of a stopping time dependent on a Poisson process and a variable $n$

Situation: We have that $\{W_t,t \geq 0\}$ is a Brownian motion and $\{N_t,t\geq 0\}$ is a Poisson process such that $N_t$ follows a Poisson distribution with parameter $t$. This process is ...
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