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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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How can an Inhomogeneous Poisson Process as a model for probability have a jump size of 1?

I have the probability of an event, happening at time $t$, $\operatorname{v}\left(t\right)$ being modelled by the following equation: $$ \tau\, \frac{{\rm d}\operatorname{v}\left(t\right)}{{\rm d}t} = ...
James Stirling's user avatar
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Expected first arrival times in poisson process.

Suppose we have two independent homogeneous Poisson processes $\text{HPP}(\lambda_1)$ and $\text{HPP}(\lambda_2)$. We denote $(N_t^1)_{t\geq 0}$ as the number of arrivals in $\text{HPP}(\lambda_1)$ ...
VlakecTomaz's user avatar
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Equivalent characterizations of a Poisson process

Consider a counting process $N(t)$, and the following three conditions that I believe are equivalent: $$ \begin{aligned} (A) \quad& N(t) \text{ is a Poisson process of intensity }λ \\ (B) \quad&...
Julius Plenz's user avatar
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2 answers
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Probability 2 earthquakes happen in a period of time.

The amount of earthquakes that happen at island X follows the Poisson process with mean 2 . Given that 2 earthquakes have happened in this year, find the probability both the earthquakes happen ...
user1259172's user avatar
3 votes
1 answer
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Interarrival Times for a Non-Homogeneous Poisson Process

It is well known that the interarrival times for a standard (i.e. homogeneous) Poisson Process follow an Exponential Distribution (What is the correct inter-arrival time distribution in a Poisson ...
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Relationship Between Poisson Process and Birth and Death Process

I am learning about Birth and Death (Stochastic) Processes (https://en.wikipedia.org/wiki/Birth%E2%80%93death_process): When a birth occurs, the process goes from state $n$ to $n + 1$. When a death ...
konofoso's user avatar
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Expectation of real number exponentiated by Poisson process

Consider a Poisson process $N_t$ with intensity $\lambda>0$ and let $x$ be a real-valued number. In principle, from the properties of the Poisson distribution, we have: $$\mathbb{E}(x^{N_t})=e^{\...
Morris Fletcher's user avatar
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Rick durrett Stochastic process - Theorem 2.12

Following is a theorem statement in Rick Durrett's book on Stochastic processes. Theorem 2.12. Suppose that in a Poisson process with rate A , we keep a point that ...
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Janossy densities and the relation with density with respect to a Poisson point process

In the spatial point process literature, it is customary to specify a point process by assigning its density with respect to a Poisson point process. For instance, an Hardcore point process $\mathbf X$...
mariob6's user avatar
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Exercise on Poisson processes

Let $(N^k)_{k\in Q}$ be a countable family of homogeneus Poisson processes with respective intensities $(\lambda_k)_{k \in Q}$. Assume $\lambda = \sum_{k \in Q}\lambda_k < \infty$. Let $T_1$ denote ...
Marco's user avatar
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Poisson process and a stopping time

Let $(N_t)$ be a Poisson process. For $a \in \mathbb{N}$, is the random variable $$\tau = \inf \lbrace t \ge 0 : N_t = a \rbrace$$ a stopping time with respect to $\left(\mathcal{F}_t^N\right)$? I ...
Barabara's user avatar
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Formal proof of joint pdf for arrival times of a Poisson process

Consider a probability space $(\Omega,\mathscr{F},\mathbb{P})$ which supports a Poisson process $N$. Let $T_1$ and $T_2$ be the first two arrival times from $N$, while $\xi_2$ is the first inter-...
Morris Fletcher's user avatar
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Derivation of Expected Number of Occurrences in a stochastic Intensity Poisson Process Using Given Axioms

I have a question concerning the expected number of occurrences in a random intensity Poisson process during a specific interval. Let $N_t(h)$ be a random variable counting the number of events that ...
MLER's user avatar
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Calculate $\mathbb E [\mathbb E[ \sum_{n=0}^{\infty}\mathbb E [T | X_T = n]\mathbb P[X_T=n] | T]]$, where $T\sim$ Exp and $(X_t)$ a Poisson process

Assume $T \sim \text{Exp}(\mu)$ and let $X = (X_t)_{t \geq 0}$ be a Poisson process of rate $\lambda$. I am trying to calculate $\mathbb E [\mathbb E[ \sum_{n=0}^{\infty}\mathbb E [T | X_T = n]\mathbb ...
hm1912's user avatar
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Find the probability that strictly $2$ people arrived in the first hour in a Poisson Process

Let the number of people arriving at a shop within the time interval $[0,t]$ be $X_t$. $4$ customers arrived in the first $2$ hours. Find the probability that strictly $2$ people arrived in the first ...
Rory-Laughlin's user avatar
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Stochastic processes with elegant closed form distributions

A Poisson process $(X_t)_{t\ge0}$ of rate $\lambda$ has the well-known closed form distribution $\mathbb{P}(X_t=k)=\frac{(\lambda t)^ke^{-\lambda t}}{k!}$. Yule processes, with rates $\lambda_n=n\...
hegash's user avatar
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Probability that a jump belongs to a certain class

Let $N^a$, $N^b$ be two jump process with stochastic intensity process $(\lambda^a_t)_{t\in\mathbb{R}}$, $(\lambda^b_t)_{t\in\mathbb{R}}$ (the lambdas are $\mathcal{F}_t$ -adapted). Let $N$ defined by ...
Sabrebar's user avatar
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Fastest growing renewal process

There is a lemma in my lecture notes stating that the renewal function $m(t)=\mathbb{E}[X_t]$, of a renewal process $X_t$ with inter-arrival density $f$, satisfies the upper bound: $$m(t)\le Ce^t \...
hegash's user avatar
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Probability distribution function that governs the number of points in histogram bins. Consistency between multiple dimensions?

Let's have a continuous random variable $x$ with some probability density function $f(x)$. A bin is given by an interval $[x_1, x_2)$. Given a total number of points $N$ generated from the PDF $f(x)$, ...
user16320's user avatar
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2 answers
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When can we compute the CDF of the Compound Poisson process

Suppose we have a Compound Poisson Process with intensity $\lambda$ $$ S_t = \sum_{i=1}^{N_t} X_i. $$ We can compute the formula for the CDF as follows \begin{align*} F_{S_t}(x) = P(...
VlakecTomaz's user avatar
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M/M/c Queue Model Solutions for Average Waiting Time and Queue Length

I am seeking assistance with a queueing theory problem involving the M/M/c queue model from my textbook. I have attempted to solve the problem and would greatly appreciate it if someone could review ...
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Questions about Probability: Theory and Examples problem 5.3.8: why do we have $\lambda<x(1-p)$

I was doing Probability: Theory and Examples problem 5.3.8 To use Theorem 5.3.8, we need to prove $\mathbb{E}_x[\phi(X_1)]<\phi(x)$, which is just $\mathbb{E}_x[X_1]<x$. After some simple ...
Ho-Oh's user avatar
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Minimum record of exponentials getting broken finitely/infinitely many times

Let us consider the following scenario: we have a sequence $(X_n)_{n\geq 1}$ of independent random variables, where for every integer $n\geq 1$, $X_n$ is exponentially distributed with parameter $\...
sicmath's user avatar
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1 answer
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Arrival of voters is modelled as a Poisson process. How to find the probability of a certain event, conditioned on the number of votes?

This is a question from Gallager's book "Stochastic Processes - Theory for Applications" Q) The voters in a given town arrive at the place of voting according to a Poisson process of rate $\...
Divyanshu Shambharkar's user avatar
1 vote
1 answer
86 views

Expected number of passengers in a bus, if both bus and passengers arrival time have a Poisson distribution.

Here's the full question In Poisson Bus City, there is a shuttle bus that goes between Stop A and Stop B, with no stops in between. The times at which the bus arrives at Stop A are a Poisson point ...
fresh_start's user avatar
1 vote
1 answer
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expected value of exponential compound poisson process

Let $Z(t)=\sum_{i=1}^{N(t)} X_i$ and let $N(t)$ be a Poisson process with parameter $\lambda$ and $X_1,X_2,\dots$ positive iid random variables with density function $f_X(x)$, independent of $N(t)$. ...
Leon's user avatar
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Customers arrive at a facility according to a Poisson process

Customers arrive at a facility according to a Poisson process $N(t)$ of rate $\lambda = 5.5$ customers/hour. Each customer is admitted to the facility with probability $p=0.6$. All customers, who are ...
waterr's user avatar
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Characterization of compensated Poisson processes

I've found the following statement (but it is rather an example) in a book that states if $X$ is a local martingale and its quadratic variation has the form $$\left[X\right]=t+cX$$ where $c$ is a ...
Kapes Mate's user avatar
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Probabilities for Pure birth Process

Consider a pure birth process starting from $X(0) = 0$ with birth parameters $\lambda_0 = 1.4$ and $\lambda_1 = 1.8$ Compute the following probabilities $\mathbb{P}(X(0.2) = 0)$ and $\mathbb{P}(X(0.2) ...
waterr's user avatar
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1 answer
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Conditional distribution of interarrival time of second arrival in Poisson process

Given a Poisson process with parameter $\lambda$, what is the probability $P(T_2 \leq 2 | N(3) = 1)$, where $T_2$ is the interarrival time between the first and second arrivals? In words, given that ...
azk55542's user avatar
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1 answer
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Expectation of thinned Poisson point process

Problem statement: Assume $N=\sum_{i=-\infty}^{\infty} \delta_{\Gamma_i}$ is a homogeneous Poisson point process (hPPP) on the real line with positive rate $\lambda$. Let $N$ be independent of $(U_n)_{...
implicati0n's user avatar
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How to calculate the expectation of this Poisson-like process?

Question: Let $\tau_i\sim \text{Exp}(\lambda_0)$ iid and $\gamma_i\sim \text{Exp}(\lambda_1)$ iid and independent of each other and set $N_t=max\{k\geq 0: \sum_{i=1}^k{(\tau_i+\gamma_i)}\leq t\}$. ...
Diplodokus's user avatar
1 vote
2 answers
77 views

What rule is used in this derivation of the interarrival time for the Poisson process?

I'm working on calculating the probability distribution of the interarrival time of the Poisson process. The method used in my textbook is very strange I don't understand how the probabilities are ...
ekke's user avatar
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Regarding Construction of a Poisson Process

I'm taking a graduate level course on Stochastic Processes and encountered the following problem in one of our assignments. $\textbf{Problem:}$ Fill in the details of the of the following construction ...
Pritam Acharya's user avatar
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1 answer
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Poisson process for expected customer arrival

Customers arrive at a service facility according to a Poisson process of rate $\lambda = 5$ customers/hour. Let N(t) be the number of customers that have arrived up to time t hours. Let $W_1,W_2,W_3,.....
waterr's user avatar
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1 answer
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Poisson Process Probabilities with given parameter

Consider a Poisson process (N(t))$_{t \geq 0}$ with $\lambda = 1.5$ compute the following a) $P(N(2) = 2; N(2.5) = 3; N(3) = 6)$ b) $P(N(2.5)*N(3)=3)$ c) $P(N(2) + N(2.5) = 0)$ d) $P(N(2) + N(2.5) = 1)...
waterr's user avatar
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Inductive proof Poisson process counts follow Poisson distribution

In converting between two definitions of a Poisson process, namely starting from the "exponential inter arrival-times" definition and attempting to prove the "Poisson distribution of ...
hegash's user avatar
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True or false: In a 2D Poisson process, for every point $P$, there exists a convex $1000$-gon with Poisson points as vertices, that contains only $P$.

I made a Desmos graph that generates $30$ uniformly random black points in a disk, with the centre of the disk in red. I asked myself, "Can I always draw a convex quadrilateral with four of the ...
Dan's user avatar
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9 votes
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A walk on a $2D$ Poisson process in which every step goes to the nearest unvisited point: expected distance from origin after $365$ steps?

Uncle's epic journey One year ago, my uncle set off from our village on an epic journey, in which every day he travels to the nearest unvisited village and stays there for the night. The villages in ...
Dan's user avatar
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2 votes
1 answer
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Higher conditional intensity implies higher variance of the resulting counting process

Suppose $N=(N_t)_{t\geq0}$ is a simple counting process that is driven by the conditional intensity process $(\lambda_t)_{t\geq0}$. That is, for $(\mathcal F_t)_{t\geq0}$ the natural filtration ...
Václav Mordvinov's user avatar
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Superposition of Poisson Processes with a weighted sum

If L(t) and M(t) are independent Poisson Processes, for which a and b is N(t) = aL(t) + bM(t) a Poisson Process. A necessary condition for it to be Poisson is if N(t) has Poisson distribution and a ...
revision's user avatar
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What is the steady state distribution of this Poisson process with non-constant rate?

I am looking for the steady state distribution of the following Poisson process: $$d x(t) = -k_1(x(t)-k_2)dt + k_3dN(t)$$ where $k_1$, $k_2$ and $k_3$ are constants and the rate $\lambda(x)$ of the ...
user1031129's user avatar
4 votes
1 answer
130 views

A square contains many random points. From each point, a disc grows until it hits the nearest neighboring point. What is the total area of the discs?

A unit square lamina contains $n$ independent uniformly random points. Each point is the centre of a disc whose perimeter touches the nearest neighboring point. Here is an example with $n=20$. In ...
Dan's user avatar
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Probability Density Function of Compound Poisson Process

I am trying to determine if it is possible to compute the probability density function (PDF) of a compound Poisson process $Y(t) = \sum_{i=0}^{N(t)} X_i$, where $N(t)$ is governed by a Poisson process ...
Josh Pilipovsky's user avatar
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1 answer
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Suposse that $\{N_t\}$ is a Poisson process with rate $\lambda>0$, and the arrival times are $S_1,S_2,\dots$.

Suposse that $\{N_t\}$ is a Poisson process with rate $\lambda>0$, and the arrival times are $S_1,S_2,\dots$. Evaluate the following in terms of $\lambda$. $(i) \, \mathbb{P}(N_1 \geq 1, N_3 \leq 2)...
Mr M's user avatar
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Is any independent counting process Poisson?

I have seen a proof for the following statement: If a counting process $\left\{ N(t) \mid t\ge 0\right\}$ is homogeneous and has independent increments, i.e., $N(b_1)-N(a_1)$, $\dots$, $N(b_n)-N(a_n)$ ...
mathslover's user avatar
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When is compound Poisson process a martingale?

here's my proof of a claim that the Compound Poisson Process (CPP for short) is a martingale $\iff$ the expected value of the iid random variables we are summing is $0$. A stochastic process $X_t$ ...
VlakecTomaz's user avatar
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1 answer
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Poisson Point Process as a random measure and finiteness of an integral

Suppose that $v$ is a Radon measure on $(0,\infty)$ and let $X$ be a Poisson Point Process on $(0,\infty)$ with intensity measure $v$. Let $Y:=\int xX(dx)$. In Theorem 24.17 of Probability Theory by A....
Enrico's user avatar
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2 votes
1 answer
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Showing $e^{\alpha N_t}W_t$ is a mixed process

On the probability space ($\Omega, \mathcal{F}, (\mathcal{F}_t), \mathbb{P})$, let $(N_t)$ be a Poisson process with intensity $\lambda$ and $(W_t)$ be a Brownian motion (the two processes being ...
Dada's user avatar
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Compound Poisson processes. Interestnig properties

I'm currently finishing my bachleor's studies and have to pick a topic for my bachelor's thesis. I'm leaning towards analysing the compound Poisson process, but since the thesis should be about 30 ...
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