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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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 $...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 \...
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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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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 $\...
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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$(\lambda)$ ...
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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 ...
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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\{...
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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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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\...
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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 ...
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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:
$$
\begin{aligned}
\mathbb{E}\left[\sum_{k=1}^{N_t}f(S_k)\right]
&= \sum_{n\ge 0}
\mathbb{E}\left[
\left.\sum_{k=1}^{n}f(S_k)\...
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Conditional probability with merged Poisson processes
Assume spam and non-spam email arriving in an inbox can be modelled as two Poisson processes $\{N_1\}$ and $\{N_2\}$ with hourly rates $\lambda$ and $\mu$ respectively.
I am asked to compute the ...
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Differential-difference equation of continuous transformation of Poisson process
Let $N(t)$ be a Poisson process with rate $\lambda$ and let
$$u(x, t) = \mathbb E [f(x + N(t))]$$ where $f:\mathbb R\to \mathbb R$ is a continuous and bounded function, $t\geq 0$, $x \in \mathbb R$. I ...
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Jumps of a compound Poisson process
If $(Z_n)_{n\in\mathbb N}$ is an i.i.d. process with values in a normed $\mathbb R$-vector space $E$, then $$W_n:=\sum_{i=1}^nZ_i$$ is called random walk with step distribution $\mathcal L(Z_1)$.
Now ...
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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 ...
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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 ...
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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)$....
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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. ...
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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 ...
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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 ...
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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. ...
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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 ...
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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 ...
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$\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,
$$
\...
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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 ...
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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 $P_0(0) = 1$. ...
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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) ...
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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 ...
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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 ...
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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 ...
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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 \...
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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]} $...
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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 ...
