# 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 random measure from a Poisson process

I would like to show that « Poisson random processes » and « Poisson random measures » are the same objects. More precisely, suppose that $N_t$ is a Poisson random process with intensity $\lambda$ on ...
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### Email arrivals in rates and time intervals according to a Poisson process

Regular (not junk) emails arrive at your inbox according to a Poisson process with rate $r$; and junk emails arrive at your inbox according to an independent Poisson process with rate $j$. Assume both ...
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### Integrating the increments of a Poisson Process [closed]

In examples 11.4.4, how exactly are we able to say that the answer to the integration will be 0? Φ(s) takes the value of either 0 or 1.
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### $M/M/2/3$ Queuing Theory Word-Problem

A service center consists of two servers, each working at an exponential rate of two services per hour. If customers arrive at a Poisson rate of three per hour, then, assuming a system capacity of at ...
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### Distribution of a poisson process in an exponential random variable

I think I already solve this problem but I would like to know your opinion about the solution. Let $\{X(t)\}_{t \geq 0}$ be a poisson process with parameter $\lambda$, and T an exponential random ...
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### Understanding $\lambda$ in the definition of Poisson distributions

I am trying to understand the meaning of $\lambda$ in Poisson distributions. I know that it is the average rate of occurrences of the event, but I have not been able to fully understand what that ...
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### Transition probabilities of a pure death process with exponential lifetime

Problem: The lifetimes of elements of a certain type are independent and exponentially distributed with parameter $\lambda > 0$. At time $t= 0$ there are $X_0=n$ living elements present. Let $X_t$ ...
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### How to simulate the distance to the nearest neighbor in a poisson process?

I am new to the topic and I have problems with the simulation of this problem: In a study, the trees in a certain region of a forest were counted, a total of 630 trees were counted, of which 91 are ...
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### Joint tails of normalized Poisson process with different intensities

Short version For those of you who don't want to read this rather long post, a condensed version of my question is: Given a Poisson process $S_u$ whose intensity $\lambda_u$ depends on some parameter ...
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### Maximising a function of $t$ for a compound Poisson Process

I was trying to solve the following question from Sheldon Ross, Introduction to Probability Models. The number of missing items in a certain location, call it $X$, is a Poisson random variable with ...
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A continuous-time Markov Chain $(X_t)_{t\ge 0}$ is called time-homogeneous if $$\qquad (*) \qquad P(X_{s+t}=j|X_s=i)=P(X_t=j|X_0=i), \forall s\ge 0$$ Strictly speaking, a Poisson process $(N_t)_{t\... • 1,922 0 votes 0 answers 29 views ### Is this a compound Poisson process? I am doing some problems related with the Poisson Process and i have a doubt on one of them. The problem is stated as follows: A doctor works in an emergency room. The emergencies arrive according a ... 1 vote 0 answers 24 views ### how to compute the expectation of the sum of waitting times of poisson process$\mathbb{E}[\sum_{k=1}^{N_t}f(S_k)]? In short , I don't understand the step 2 of the answer: \begin{aligned} \mathbb{E}\left[\sum_{k=1}^{N_t}f(S_k)\right] &= \sum_{n\ge 0} \mathbb{E}\left[ \left.\sum_{k=1}^{n}f(S_k)\... • 507 0 votes 0 answers 14 views ### Conditional probability with merged Poisson processes Assume spam and non-spam email arriving in an inbox can be modelled as two Poisson processes \{N_1\} and \{N_2\} with hourly rates \lambda and \mu respectively. I am asked to compute the ... 1 vote 0 answers 27 views ### Differential-difference equation of continuous transformation of Poisson process Let N(t) be a Poisson process with rate \lambda and letu(x, t) = \mathbb E [f(x + N(t))]$$where f:\mathbb R\to \mathbb R is a continuous and bounded function, t\geq 0, x \in \mathbb R. I ... -1 votes 1 answer 34 views ### Jumps of a compound Poisson process If (Z_n)_{n\in\mathbb N} is an i.i.d. process with values in a normed \mathbb R-vector space E, then$$W_n:=\sum_{i=1}^nZ_i$$is called random walk with step distribution \mathcal L(Z_1). Now ... • 12.8k 0 votes 1 answer 23 views ### Joint pdf of Poisson processes I recently stumbled upon this question: Let T1 and T2 be the times for the first and second jump of a Poisson process with rate λ > 0. Find the joint probability density function of T1 and T2 After ... • 55 3 votes 2 answers 150 views ### Doubts about Proof of Durrett Theorem 3.7.4. Thinning of Poisson Process I am having trouble understanding Durrett's logic in his proof of the thinning of the Poisson process. Here is the statement of the Theorem: N_j(t) are independent rate \lambda P(Y_i = j) Poisson ... • 459 0 votes 1 answer 23 views ### Find posterior distribution of Poisson process knowing that the prior is Exponential(1). This is the problem: Bus arrival times form a Poisson process with intensity 𝜆 measured in buses per hour. Your prior distribution on 𝜆 is that 𝜆 is an exponential random variable, Exponential(1).... 3 votes 0 answers 26 views ### Probability that two earth quakes are spaced at least 5 years between each other The problem is stated as: Alaska has over half of all earthquakes in the United States. In particular, earthquakes with magnitude>8 on the Richter scale occur in Alaska on average every 13 years. ... • 1,450 0 votes 1 answer 19 views ### Poisson Process with Random Variables I am dealing with the following problem. Problem Definitions Problem itself I converted P( N2 + 2N5 = 5 ) to P( N2 + 2(N5 - N2 + N2 ) = 5 ), which is equal to P( 3N2 + 2(N5 - N2) = 5 ). Since N2 is ... -1 votes 1 answer 59 views ### Average number of organisms at any point in time given birth and death rates. [closed] There is a population of organisms that get birthed at rate \lambda and once they are born, they die at rate \mu. Once steady state is achieved, what will the average number of organisms at some ... • 5,459 2 votes 1 answer 41 views ### A Poisson Process occurring before an event The Geophysics building at the University of Northern California is scheduled to be seismically reinforced. The reinforcement will occur at a random time uniformly distributed in the next 3 years. ... • 1,131 2 votes 1 answer 45 views ### Equivalence Definitions for Nonhomogeneous Poisson Process By Stocastic Processes, Sheldon M. Ross, The Second Edition, p.78, the definition of nonhomogeneous Poisson process is given by: The counting process \{N(t),t\geq 0\} is said to be a nonstationary ... • 351 0 votes 0 answers 10 views ### Conjugate prior for estimating one "component" of a merged Poisson process. Given Information: Let N_1(t) and N_2(t) be two independent Poisson process with arrival rate \lambda_1 and \lambda_2, respectively. Let N(t):=N_1(t)+N_2(t) denote the merged Poisson process ... 1 vote 0 answers 30 views ### \mathbb{E} \left(|N| \choose k \right) at a given time t when N is a unit rate Poisson Point Process, for a fixed positive integer k Would like someone to double check my work here. Thank you :) |N| is the number of points of the unit rate Poisson point process up to time t. At time t, |N| \sim Poisson(t). Therefore,$$ \... • 654 2 votes 0 answers 19 views ### How is this Poisson representation formula for continuous time Markov chains proved? My textbook claims the following fact. As far as I can tell it provides neither a proof nor a citation. Fact: LetS\subseteq\mathbb Z^d$be infinite. Let$\zeta_1,\cdots,\zeta_n\in\mathbb Z^dbe ... • 654 2 votes 1 answer 135 views ### pure Poisson birth process ordinary differential equations Consider the pure Poisson process defined by \begin{align} P_n'(t) &= -\lambda_n P_n(t) + \lambda_{n-1}P_{n-1}, \quad n \geq 1,\\ P'_0(t) &= -\lambda_0 P_0(t). \end{align} withP_0(0) = 1$. ... • 1,131 0 votes 0 answers 100 views ### How to solve this probability problem? Conduct an experiment that marks a random number of points on a line. The number of points marked on any line segment of length x is a random variable that follows the Poisson(ρx) ... 0 votes 0 answers 14 views ### On the continuity of the product of a left-continuous and a right-continuous stochastic process We introduce a probability space$(\Omega,\mathscr{F},\mathbb{P})$equipped with a filtration$\mathbb{F}$. We define a Poisson process$N$with intensity$\lambda$adapted to$\mathbb{F}$, and we ... 0 votes 0 answers 25 views ### Variance of Integral of Poisson process Let$\{X(t):t\ge 0\} $be a Poisson process with rate$\lambda$. T is a positive constant. Define$\xi_T=T^{-1}\int_0^TX(t)dt$. Calculate$Var[\xi_T]$. In the beginning, I want to solve it by ... 2 votes 1 answer 41 views ### Question on the proof that jump times of a Poisson process are totally inaccessible I am reading the proof that the jump times of a Poisson process are totally inaccessible from the following post : https://almostsuremath.com/2010/06/24/poisson-processes/#scn_pp_def1 From definition ... • 5,723 2 votes 1 answer 70 views ### Spatial Poisson Process on a square Can I ask for advice on proceeding with this question? It is Problem 5.5.3 (page 282) of Samuel Karlin, Mark A. Pinsky's ''An Introduction to Stochastic Modelling'' 4th edition. Let$\{X(A) : A \...
In my notes, one definition of poisson process is: $N_t$ is a poisson process of rate $\lambda$ if i) $N_0=0$; ii) for disjoint intervals $(s_i,t_i)$, $N_{(s_i,t_i]}$ are independent; iii) $N_{(s,t]}$...
My doubt arose when I wanted to add two independent exponential interarrival times of parameter $\lambda$ and $\gamma$ respectively, with $\gamma \neq \lambda$. I know that the sum of two independent ...