Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

0
votes
0answers
16 views

Waiting time in Poisson process

Let $\{X(t) : t \geq 0\}$ be a Poisson process with rate $\lambda$, and let $W_n$ denote the waiting time for the $n$-th event. For $s \geq 0$, determine $P( W_{X(t)} \leq t+s)$ and $P( W_{X(t)+2} \...
0
votes
0answers
14 views

nearest neighbor for a Poisson process in the plane

I do not know why the term called "nearest neighbour" while in all the place treat it as all the neighbours within a certain area like here Doesn't $P(X<r)=1-P(X>r)=1-P(U_r=0)$ mean the ...
1
vote
0answers
9 views

superposition of infinitely many poisson processes

I know that the superposition of two Poisson process with rates $\lambda_1$ and $\lambda_2$ is again a Poisson process with rate $\lambda_1+\lambda_2$. Thus this process has interarrival times ...
0
votes
0answers
8 views

Poisson Process/Inter-Arrival Time Question

This question is, "People arrive at a train station in accordance with a Poisson process with rate lambda. At time 0, the train station is empty. At time 10, the bus departs. Calculate the expected ...
0
votes
0answers
8 views

Show that the number of elements in a given set is a Poisson process.

Let $\quad \{ \nu_n, \xi_k^n \}\in \mathbb N$ be a family of random variables. $\quad \nu_n \sim Poiss(\lambda), \quad \xi_k^n \sim Unif(n-1,n].$ Let $\quad N_t=|\{ (k,n): k \le \nu_n, \quad \xi_k^...
0
votes
0answers
14 views

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 ...
3
votes
0answers
67 views

Double integral over square converging to $0$

I am struggling to solve the following problem as part of a bigger project that I am working on. Let $\mathcal{S} \subset \mathbb{R}^2$ be a square of length $\sqrt{n}$ centered at the origin, $f:\...
0
votes
0answers
21 views

Byzantine Fault Tolerance Threshold of Bitcoin: 1/2 or 1/3?

Although, this question is related to the Bitcoin network; however, its calculation is more relative to the Mathematics. So, let's bring it up here: According to this answer: https://bitcoin....
1
vote
1answer
34 views

Poisson process that degrades uniformly

Given that people arrive according to a poisson process with rate 4 per minute and stay for $X$~Unif[0,10] number of minutes independently of the arrival times. What is the mean and variance of the ...
0
votes
0answers
8 views

Rate parameter in Poisson process - confusion regarding terminology

I am trying to understand a derivation of maximum likelihood estimation of intrinsic dimension from a Poisson point process given here https://wiki.math.uwaterloo.ca/statwiki/index.php?title=...
1
vote
0answers
25 views

The expected distance from a point to its neighbors in homogeneous spatial poisson process

Let $n$ point are distributed as per a homogeneous spatial Poisson process of rate $λ$ in a square of side $2a$, and $R$ be the distance from a point to its neighbors within a distance $r$. What is ...
0
votes
0answers
33 views

How to calculate realization of a random process?

I am newly learner of subject of stochastic processes and my mind is full of questions. Hopefully I can ask one of them in a correct way. Suppose that X is an random variable which follows ...
0
votes
0answers
39 views

If $N$ is a Poisson process, then $\operatorname P\left[\exists t>0:N_t=n\right]=1$

Let $(N_t)_{t\ge0}$ be a Poisson process with parameter $\lambda\in[0,\infty)$ and $n\in\mathbb N$. How can we show that $\operatorname P\left[\exists t>0:N_t=n\right]=1$? Assume for the moment ...
1
vote
1answer
33 views

The average time before a person find their group

Imagine there are $N$ people throwing a party. For any two of them, the time before they meet each other and stick together thereafter is independent, and obeys an exponential distribution whose $\...
0
votes
0answers
21 views

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 ...
1
vote
2answers
63 views

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 ...
0
votes
0answers
16 views

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?
0
votes
0answers
8 views

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?
0
votes
0answers
15 views

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 ...
-1
votes
0answers
18 views

Construction of probability measure for Hawkes process

How to construct a probability measure for the Hawkes process? Like here
1
vote
0answers
29 views

Distribution of inter-arrival time of non-homogenous Poisson Process

What is the distribution of inter-arrival time of a non-homogenous Poisson Process? In other words, if $T_n$ is a non-homogeneous Poisson process with intensity $\lambda(s)$, and $S_{n+1} = T_{n+1} - ...
0
votes
0answers
27 views

Conditional law of Poisson process

1)What does the notation for $G_n$ and $H_n$ mean? 2)How would $G_n$ and $H_n$ look like for a Poisson process? 3)How to show that $v$ would be the compensator of a Poisson process? Thanks!
2
votes
0answers
18 views

Counting process with independent, stationary increments is Poisson

Suppose that $L_t$ is a counting process, i.e. $$ L_t= \sum_{i \in \mathbb N} 1_{T_i \le t} $$ for a collection $(T_i) _{i \in \mathbb N}$ of stopping times. I have often seen the claim that if $L_t$...
1
vote
2answers
76 views

Interarrival Time Distribution of a Poisson Process

For a Poisson Process with parameter $\lambda$ restricted to the interval $[0, 1]$, what is the probability that at least one of the interarrival times (including the time between $0$ and the first ...
0
votes
0answers
15 views

proof of the expected value for two Poisson point process

Consider $H_1$ and $H_2$ as two independent homogeneous Poisson point process in $\mathbb{R}^2$ space of intensities $\lambda_0$ and $\lambda_1$, respectively. We divid a square area into some region ...
0
votes
0answers
12 views

Counting Process Conditioned on Random Rate

Having some trouble figuring out this problem. Say you have a counting process $N_t$ where the rate is a random variable $\Lambda$ which is exponentially distributed with rate $\alpha$. I'm trying to ...
1
vote
1answer
22 views

Poisson Process Conditional Expectation

Given $X_t$ a Poisson process such that $\lambda = 1$ find $E[X_1\mid X_2]$ and $E[X_2\mid X_1]$. The first one is pretty straight forward since we have $E[X_2 - X_1] = E[X_1] = 1$ so then we get $E[...
0
votes
0answers
17 views

Escaping probability for a poisson random walk

Poisson random walk: Let independent random variables $Z_i \sim Pois(\lambda)$. Consider random walk $ S_n = \sum_{i=1}^{n}X_i, $ where $$ X_i = \begin{cases} Z_i &\text{w.p}\; p\\ -Z_i &\...
3
votes
1answer
104 views

Poisson point process in 2D with reflecting boundaries

Consider a point process $\{(X_n,T_n)\}$ on a plane $[0,1/\lambda]\times\mathbf R^+$, generated from a Poisson point process $\{T_n\}$ with rate $\lambda$ on $\mathbf R^+$ (i.e. $(T_n-T_{n-1})$ is iid ...
1
vote
1answer
18 views

Why is this inequality true in proof of strong law of large numbers for renewal processes

Let $\{N_t\}_{t\ge 0}$ a renewal process with intensity $\lambda>0$. Then $\lim_{t\to\infty} \frac{N_t}{t}=\lambda$ a.s. Here $W_i$ denotes the waiting time of $N_t$ and $T_n:=\sum_{i=1}^n W_i$ ...
0
votes
1answer
24 views

Conditional Expectation of Poisson Process Interarrival Events

I'm having some trouble with something my professor said would be on our exam. For a Poisson process $N_t$ with interarrival times $X_i$, how is it that you find $E(X_i | N_t = n)$ (assuming $n\ge i$)?...
1
vote
1answer
70 views

Poisson Point Process On Real Number Line

Let $\mathbf{\Pi}$ be a homogeneous Poisson point process on the real number line with intensity $\lambda$. Let $r^{+}$ denote the distance from the origin to the closest point of $\mathbf{\Pi}$ on ...
0
votes
1answer
56 views

The sum of the total number of service representatives who spoke with all the customers who called during a minute.

I trying to solve this problem, but I can not tell if what I am doing is right, At some company, the customer support department gets called by $5$ customers per minute independently. The ...
-1
votes
1answer
43 views

If the inter-arrival times of customers are i.i.d. exponential distribution, is it necessary that the number of customers is a Poisson process?

Suppose customers arrive with time interval $U_i$ i.i.d. $Exp(\lambda)$, therefore, $$F(U_i\le t)=1-e^{-\lambda t}$$ The arrival time of customer $i$ is $$T_i=\sum^i_{j=1}{U_j}$$ The number of ...
-1
votes
1answer
32 views

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 ...
0
votes
0answers
19 views

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 $(a/2,a/2)$, $(-a/2,a/2)$, $(a/2,-a/2)$ ...
0
votes
0answers
14 views

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 $\mathbb{R}_+$...
0
votes
0answers
7 views

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{...
1
vote
2answers
51 views

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 ...
2
votes
2answers
78 views

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 $t$ ...
0
votes
1answer
33 views

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 $10$ ...
0
votes
1answer
31 views

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\\ &...
0
votes
0answers
62 views

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 ...
0
votes
1answer
57 views

How to solve this problem with Poisson distribution

Problem: A store owner observes that there are $3$ (in average) customers visiting the store per hour. He wants to find the probability that there are at least $1$ customer visiting his store in $...
1
vote
1answer
72 views

Expected value of Brownian motion at a time decided by a rate one Poisson process.

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 ...
1
vote
2answers
34 views

What is the limiting probability that there are n people in the facility?

Customers arrive at a store at a Poisson rate of λ and there is a single server with rate μ. The arrival and service times are independent random variables. Customers leave the facility immediately ...
0
votes
1answer
22 views

independent Poisson processes probability

I have been given this question to solve The numbers of claims to an insurance company from smokers and nonsmokers follow independent Poisson processes. On average 4 claims from nonsmokers and 6 ...
1
vote
1answer
16 views

Poisson process: finding probability of 1 count in an interval given that 0 counts happen in a subinterval

This was in my exam today and I'm not sure what's the correct answer. Let's say that the number of people that enter into a store in the interval $(0,t]$ (in hours) is a Poisson process where $30$ ...
0
votes
1answer
32 views

A question about Poisson Process: operating events from different sample spaces?

The following proof, that how to derive Poisson Distribution from a Poisson Process, is from my textbook, Elementry Probability Theory(Fourth Edition), written by Kai Lai Chung, Farid AitSahlia. ...
0
votes
0answers
39 views

Question related to Poisson process

Suppose busses arrive at a bus stop either with an inter-arrival time of exactly 1 min or with an inter-arrival time of exactly 10 mins. Suppose the 1 min inter-arrival times occur with probability 2/...