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.

Filter by
Sorted by
Tagged with
0 votes
1 answer
36 views

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 ...
user avatar
  • 618
0 votes
0 answers
97 views

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 ...
user avatar
0 votes
0 answers
18 views

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.
user avatar
1 vote
1 answer
26 views

$M/M/2/3$ 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 ...
user avatar
  • 11
0 votes
0 answers
18 views

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 ...
user avatar
  • 197
0 votes
0 answers
22 views

Nonhomogeneous (nonstationary) Poisson processes and splitting them

What I have learned from my recent readings is that if $\{N_t,t\geq 0\}$ is nonhomogeneous Poisson process (in $\mathbb{R}_+$) with intensity function $\lambda(t)$, and if an event occurring at time $...
user avatar
  • 408
0 votes
0 answers
29 views

Optimal strategy to win this football match

You're the trainer of a football team playing an opponent, and you can let your team play either in a defensive tactic, or an attacking one. You can switch between tactics at any moment, as often as ...
user avatar
1 vote
0 answers
39 views

Prove sum of two independent Poisson processes is another Poisson process

I was trying to prove that the sum of two independent Poisson processes is another Poisson process. I know how to prove that the sum of the Poisson distributions is another Poisson distribution. But I ...
user avatar
2 votes
1 answer
73 views

Necessary and sufficient condition for random sum of independent RVs to be a martingale

Let $M$ be a Poisson random measure on $(0,\infty)$ with intensity $\lambda dt$, where $\lambda\in(0,\infty)$. Let $(Y_n)_{n\in\mathbb{N}}$ be a sequence of independent random variables, independent ...
user avatar
0 votes
0 answers
16 views

Stationary and independent increments in a conditional Poisson process

I have a conditional Poisson process, for which each of the chunks with a given lenght have 0.5 probability of occur according to a Poisson process with rate 3 per time unit, and another 0.5 ...
user avatar
1 vote
0 answers
21 views

Counterexample for Mecke equation in higher dimensions

I am currently reading the book Lectures on the Poisson Process by Gunter Last and Mathew Penrose. (The book can be found here.) I have a question about an exercise in the book's 4th chapter (Exercise ...
user avatar
0 votes
1 answer
18 views

Probabilities involving the superposition of poisson processes

Say customers arrive at an ice cream shop at an average rate of 100 customers per day. Independently, the ice cream shop receives a restock of their ice cream at an average rate of one shipment every ...
user avatar
1 vote
1 answer
32 views

Moment Generating Function of a summation of random variables where the upper limit is also random

How do we compute the Moment Generating Function of Q(t) here? I understand that we can use Wald's Equation to compute E[Q(t)]. Is there any theorem which can help me solve for the Moment Generating ...
user avatar
0 votes
0 answers
23 views

Let {Xt}t≥0 a compound Poisson process with λ=2 where Yi∼Exp(1). We define τ=inf{t≥0:Xt≥10}. Find the PDF of τ.

Hello I have been aked this and I don't know how to proceed: Let {Xt}t≥0 a compound Poisson process with λ=2 where Yi∼Exp(1). We define τ=inf{t≥0:Xt≥10}. Find the PDF of τ. I would be great if you ...
user avatar
0 votes
0 answers
19 views

Concentration of the Poisson Process around its mean

Let $P$ be a Poisson process with parameter $\lambda$. The goal is to show that $$ \lim _{n \rightarrow \infty} \sup _{t \leq 1}\left|\frac{P_{n t}}{n}-\lambda t\right|=0, \quad \text { a.s. } $$ This ...
user avatar
1 vote
1 answer
18 views

is every counting Process a non homogeneous Poisson process?

I'm working on a counting process and trying to prove that this counting process is a Poisson point process (non homogeneous). I have the 3 conditions : $N(t)>0$ $N(t)\in \mathbb N,\forall t \in \...
user avatar
3 votes
1 answer
65 views

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 ...
user avatar
  • 764
0 votes
0 answers
34 views

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 $X_t$ ...
user avatar
0 votes
0 answers
16 views

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 ...
user avatar
0 votes
0 answers
33 views

Renewal process with inter-arrival time distributed as gamma: Model estimation

Let's start with the Poisson process: If $N_t$ is a Poisson process with parameter $\lambda$, then we know that the inter-arrival time distribution is an exponential distribution with parameter $\...
user avatar
0 votes
1 answer
31 views

Showing it's a Poisson Process

I have a question when reading Essentials of Stochastic Process by Richard Durrett, 2.2.1 Constructing the Poisson Process. It says, Let $\tau_1,\tau_2,\dotsc$ be independent exponential$(\lambda)$ ...
user avatar
0 votes
0 answers
17 views

Distribution of doubly stochastic Poisson or Cox process

This is an exercise in Stochastic Processes with Applications (Bhattacharya and Waymire) IV.1.6: Suppose that the parameter $\lambda$ (mean rate) of a homogeneous Poisson process $\{X_t\}$ is random ...
user avatar
1 vote
1 answer
45 views

Number of events in any interval for a Poisson process

Let $\{N(t),t\geq 0\}$ be a homogeneous Poisson process with rate $\lambda$. $N(t)$ is defined to be the number of events in $(0,t]$. Since $N(0)=0$ by definition, we can conclude that $P\{N(t)=i\}=P\{...
user avatar
  • 408
0 votes
0 answers
5 views

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 ...
user avatar
0 votes
0 answers
23 views

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 ...
user avatar
0 votes
1 answer
20 views

How to make a Poisson process time-homogeneous?

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\...
user avatar
  • 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 ...
user avatar
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)\...
user avatar
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 ...
user avatar
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 let $$u(x, t) = \mathbb E [f(x + N(t))]$$ where $f:\mathbb R\to \mathbb R$ is a continuous and bounded function, $t\geq 0$, $x \in \mathbb R$. I ...
user avatar
-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 ...
user avatar
  • 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 ...
user avatar
  • 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 ...
user avatar
  • 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$(1)$....
user avatar
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. ...
user avatar
  • 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 ...
user avatar
-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 ...
user avatar
  • 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. ...
user avatar
  • 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 ...
user avatar
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 ...
user avatar
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, $$ \...
user avatar
  • 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: Let $S\subseteq\mathbb Z^d$ be infinite. Let $\zeta_1,\cdots,\zeta_n\in\mathbb Z^d$ be ...
user avatar
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} with $P_0(0) = 1$. ...
user avatar
  • 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) ...
user avatar
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 ...
user avatar
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 ...
user avatar
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 ...
user avatar
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 \...
user avatar
0 votes
1 answer
38 views

Superposition of Poisson Process is still Poisson Process

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]} $...
user avatar
  • 113
0 votes
0 answers
12 views

If I have hypoexponential interarrival times, then does it fulfill a poisson process?

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 ...
user avatar

1
2 3 4 5
26