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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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 ...
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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 ...
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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) ...
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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 ...
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Distance distribution of poisson points to nearest shape boundaries. [closed]

I am trying to model the distribution of distances from Poisson-point generated points to the nearest boundary/edge of shapes that have also been distributed in a Poisson-point process. In this case, ...
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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)_{...
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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 $N_t=max\{k\geq 0: \sum_{i=1}^k{(\tau_i+\gamma_i)}\leq t\}$. ...
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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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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,.....
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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)...
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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 ...
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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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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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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 ...
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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
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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 ...
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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)...
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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., $N(b_1)-N(a_1)$, $\dots$, $N(b_n)-N(a_n)$ ...
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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 $X_t$ ...
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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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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 ...
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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 ...
VlakecTomaz's user avatar
6 votes
2 answers
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Insurance company with claims following a Poisson Process. Calculate the probability that the capital is always positive throughout the first 4 days.

Suppose that claims are made to an insurance company according to a Poisson process with rate $10$ per day. The amount of a claim is a random variable that has an exponential distribution with mean $1,...
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Spatial Poisson process of discs and the $r \rightarrow 0$ limit

Consider a (homogeneous) spatial Poisson process $\Pi$ in $[0,1]^2$ with constant rate $\lambda^2$. For each $x \in \Pi$, let $D_x$ be the disc centered at $x$ with radius $r$, and let $D = \bigcup_{x ...
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Density of $T_2 - T_1$ of a non homogeneous poisson process with $\lambda$(t) = $\frac{1}{1 + t}$ intensity

Me and my teammate have this question to solve in our homework. The question is the following: A N(t) non homogeneous Poisson process with intensity $\lambda$(t) = $\frac{1}{1 + t}$. Find the density ...
Samuel Fournier's user avatar
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Central limit theorem for inhomogeneous Poisson process

I consider $N_t$ an inhomogeneous Poisson process, especially $N_t$ follows a Poisson law with parameter $\int_{0}^{t}\lambda(s)ds$. We assume that $\frac{1}{t}\int_{0}^{t}\lambda(s)ds\to \sigma^{2}$ ...
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How can I check that the following process is a martingale?

Let $(\Omega, \mathcal{F}, (\mathcal{F}_t), \Bbb{P})$ be a filtered probability space. Net $N$ be a poisson process with parameter $\lambda>0$. Let $h$ be a bounded measurable function and define $...
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How can I show that a Poisson process with my definition below has stationary and independent increments?

We had the following definition: Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_t, \Bbb{P})$ be a filtered probability space. An $(\mathcal{F}_t)_t$ Poisson process $(N_t)_{t\geq 0}$ is a right ...
Summerday's user avatar
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$M/G/\infty$: application of marking and transformation, finding the mean measure

Consider a queue $M/G/\infty$, starting with arrival time of calls as a PPP$(\Lambda)$ and lengths of calls as $iid$ random variables with common distribution $G$. The times when the calls terminate ...
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Understanding Sojourn times of M/D/1 queue

I am trying to understand how to approach a problem involving a Poisson Process queue with a deterministic service time. We have that the mean rate of arrival time is your standard $\lambda$ customers ...
IPreferAlgebra's user avatar
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2 answers
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How to calculate variance in a Poisson process with exponential lifetimes of arrivals?

I am trying to understand how to combine the concepts of Poisson process and the birth and death process. I have a Poisson process where people arrive with rate $\lambda$ -- so when an event occurs, ...
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Relative reduction in $n$th lowest record value follows a uniform-[0,1] distribution

Let $U_{1},U_{2},U_{3},\ldots$ be a sequence of i.i.d. uniform-[0,1] random variables. Define the lower record times as: \begin{equation*} \sigma_{n+1} =\min{\{k\mid U_{k}<U_{\sigma_{n}}\}}, \end{...
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How to use the poisson process for BLE neighbor discovery?

I am trying to use the Poisson process for BLE(Blueetooth Low Energy) neighbor discovery. In this context, there are two devices called Advertiser and Scanner. The advertiser and scanner emit and ...
Nebex Elias's user avatar
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Distribution of waiting time conditioned on a fixed time length

Suppose, I work in a factory production line. The time for me to finish wrapping product A (or B) is exponentially distributed with parameter $\lambda_1$ (or $\lambda_2$), i.e., $X\sim\exp(\lambda_1)$,...
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Point removal from a Poisson point process

Let $\pi$ be a point process on $\mathbb R$ with intensity $\lambda$ (Lebesgue measure). We start at $Y_0=0$ and perform the following algorithm : We select the point closest to $0$ and call it $Y_1$, ...
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Supremum of a Poisson process

Let $(P(t))_{t\geq0}$ be a Poisson process of parameter $1$ (for instance, let $P(t) = \mathrm{Card}\{k \in \mathbb{N}^*: S_k \leq t \}$ where $S_k = E_1+\ldots+E_k$ is a sum of $k$ i.i.d exponential ...
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How do I find the average distance between a point and its nearest neighbor given the points are generated by a poisson process.

Given I know the area of the space that points can be in, the total number of points, and that a Poisson process generated the points, how can I find the average distance between any given point and ...
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Poisson process with exponential translations

Let $N(t)$ be a standard Poisson process with intensity $\lambda>0$. As we know, for $n \geq 1$ we can define the time of $n$th event as $ S_n = \sum_{k=1}^n T_k, \; S_0 = 0, $ where $(T_k)$ are ...
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Joint expectation in Poisson process

Let $\{N(t);t ≥ 0\}$ be a Poisson process with intensity $λ > 0$. We must compute $E(N(1)^2 \cdot N(2))$ without any other information. Because it is a Poisson process then it has independent ...
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Poisson distribution and the choice of unit

Suppose that the average number of cars passing by a building, $X$, is 3 cars per minute. Assuming that $X$ follows the Poisson distribution with mean $\lambda=3$, the probability that 1 car passes ...
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2 votes
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PDF for total duration of "on" state in a Poisson process with binary values

I have a process where, at times given by a Poisson law of rate $\lambda$, a system picks two values - say "on" or "off" - with respective probabilities $p_0$ and $1 - p_0$. What ...
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Calculate expected value, covariance and conditional expectation of a Poisson process

$ N=N(t) $ Poisson process with intensity ( \lambda>0 ). (a) $ X=\{X(t), t \geqslant 0\} $ given by $ X(t)=N(t)-t \lambda , t \geqslant 0 $ Calculate $EX(t)$ and $ {Cov}(X(s), X(t)) \quad \forall s,...
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o(h) properties within a poisson processes

I have this as part of a theorem for the sum of two poisson processes being itself a poisson process. I don't particularly understand why the fourth line equals the last line of the theorem. I believe ...
loaf's user avatar
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Explanation of notation on Levy-Ito decomposition in a paper by El Fatini and Boukanjime?

In El Fatini and Boukanjime "Stochastic analysis of a two delayed epidemic model incorporating Lévy processes with a general non-linear transmission" paper, the first equation of (3) is ...
Math's user avatar
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Expectation of sde, need to calculate the probabilty of summation of gamma distribution.

Suppose $z(t)$ is the solution of sde. Let, $ \widetilde{z}(t) = z(t) - z(t_{2k})\textbf{1}_{k\neq 0}$ for $t_{2k}\leq t < t_{2k+1}$, $ \widetilde{z}(t) = 0$ for $t_{2k+1}\leq t < t_{2k+2}$ for $...
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Unnormalized Arrival Times of Nonhomogeneous Poisson Process

Consider a nonhomogeneous Poisson process $N(t)$ for $t\ge0$ defined by the instantaneous intensity $\lambda(t)\ge0$ and mean value function $\Lambda(t)=\int_0^t\lambda(s)ds$. We know that $$ \mathbb{...
Andras Vanyolos's user avatar
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Generalization of Campbell's theorem

Consider a point process $\Phi=(X_i)$ defined on $[0,1]$. Is there any formula for computing $$ \mathbb{E}\left[\frac{\sum_{X_i\in \Phi}X_i^d}{(\sum_{X_i\in \Phi}X_i)^d}\right]? $$ Campbell’s ...
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