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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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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{...
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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 ...
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Variance decomposition of Poisson process

Let $N(t)$ a poisson process with intensity $\lambda$ and $X$ a positive random variable independent of $N$. Let $f$ a real valued increasing function with $f(0)=0$. Consider $C(t):=N(Xf(t))$. ...
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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 $t$ ...
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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 $10$ ...
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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\\ &...
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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 ...
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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 $...
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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 ...
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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 ...
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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 ...
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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 $30$ ...
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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. ...
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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/...
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Poisson Process expectation of time of an event given number of events until that time shows Uniform distribution characteristics

Question: On a weekday, buses arrive to a certain stop with respect to a Poisson process with rate $\lambda = 2$ per hour. A regular weekday is assumed to start at 6:00 AM. Given that exactly $10$ ...
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Poisson Event After Time Interval

So I have this Poisson Problem that I'm struggling with, and the basis is that you have a server that fails once every four hours (so the average is 1/4 of a crash per hour). The question that I'm ...
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Joint distribution of Poisson process

$\Pi(t)$ is a Poisson process I want to calculate joint distribution of $(\Pi_{t_1}\Pi_{t_2}...\Pi_{t_n})$ Please check my solution: Lets define random variables $X_1 = \Pi_{t_1}, X_2 = \Pi_{t_2}- ...
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How can you prove time between random Poisson points is exponential?

How can you prove time between Random Poisson Points is exponential? I know you can derive the Poisson Distribution from exponential but i'm not sure if this is enough to prove about the time ...
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2 Poisson distributions time distribution given one of them occurs first

Suppose that A and B are two independent Poisson distribution with parameter $\lambda_a$ and $\lambda_b$ denoting the number of, say the received emails per hour in two different computers. B is ...
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Interarrival Times of Poisson Process

Let $(N_s)$ be a homogene Poisson Process with rate $\lambda>0$ and let $t>0$ be fixed. $T_{N_t}$ is then the last arrival time before time $t$ and $T_{N_t+1}$ is the first arrival time after ...
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Random time change for a Poisson process and convergence with respect to the Skorohod topology

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\left(Y^{(n)}_k\right)_{k\in\mathbb N_0}$ be a time-homogeneous Markov chain on $(\Omega,\mathcal A,\operatorname P)$ and $$X^{(n)}_t=...
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Steps of a Markov chain subordinated to a Poisson process

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\left(Y^{(n)}_k\right)_{k\in\mathbb N_0}$ be a time-homogeneous Markov chain on $(\Omega,\mathcal A,\operatorname P)$ $D([0,1]):=\...
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Sample from Poisson process with no collisions.

On a 2D square $[0,\ 1]^2$, I can draw a random configuration of points $c = (n,\ (x_i)_{1..n})$ with interesting independence properties using a Poisson point process: draw the number of points $n \...
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Properties of Poisson Processes Further Investigated

I am reading Introduction to Probability Models by Sheldon M. Ross, and I am having a difficult time comprehending this example. The text explains this section 'Further Properties of Poisson Processes'...
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Birth-death process Expected waiting time in when in invariant distribution

Say I have a birth-death process, with birth having Poisson distribution with parameter $\lambda$ and death having poisson distribution $\mu$. Assuming that both stochastic processes, birth and death, ...
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Calculating necessary assumptions on simple Poisson process

I'm reading a textbook on probability theory, and the author has recently introduced the Poisson process. We're counting occurrences in an interval of time; this quantity is a random variable for each ...
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birth-death process in a two state system

I'm trying to understand a stochastic process, and i'm not sure about a system that i'm studying. Consider a two state system with a certain number $N$ of time-depending variables $x_i$ that can ...
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Pick Balls with Unequal Probabilities

There are some balls with 4 different colors in a black box. The ratio of A color ball is 10%, B is 20%, C is 30% and D is 40%. The rule of this game is that you randomly pick one ball and mark the ...
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Number of people waiting in a M/G/$\infty$ queue at time t

For a M/G/$\infty$ queue (parameter of M is $\lambda=1$), we are given that G ~ U(0,3). My goal is to find the probability that there is no one waiting in the queue at time t = 10. So far, I have ...
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Poisson and convergence in distribution

Let $X_1,X_2,...$ be iid with mean $\mu$ and variance $\sigma^2$. Let $N_{\lambda}$ be poisson($\lambda$) independent of the $X_i$'s. (a) Find the limit in distribution as $\lambda \rightarrow \infty$...
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Poisson process to Bernoulli process

I have three jobs $A,B,C$ with average arrival rates $a,b,c$. Each is independent poisson process. System waits until $10$ jobs arrive and then buffers them. Determine the probability that exactly two ...
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Independently Marked Point Process - Void Probability

In the field of wireless communication Marked Point Process plays a vital role. For example, consider that the wireless nodes are uniformly distributed in a $\mathbf{R}^2$ around the origin(...
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What is the relation between markov process , markov chain and poisson process?

I am unable to understand these concepts clearly. I am getting lost while reading about these. I have googled about these concepts but still nothing helped me to understand these concepts. according ...
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Probability of Arrivals at a bank

Suppose that the probability that a given customer entering a bank is between 50 and 70 years of age is 5/9. a.) On a given day, compute the probability that the 7th customer entering the bank is ...
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Poisson process detected with prohability p

I have a question about a homework exercise of my statistics class, concerning poisson processes: Consider a poisson process with rate $\lambda$. An event is detected only with the prohability $p$....
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Probability of an event with multiple conditions

Could you inform me please, how can I calculate conditioned probability of several events? I have 3 events A, B, C; I know P(B|C) and I want calculate P(A|B,C). Is it possible? In the special case B,...
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Poisson process — conditional expectation problem

Let ${N_t}$ be homogeneous Poisson process with rate $\lambda$. The random variables $$X_i = N_{t_i} - N_{t_{i-1}}$$ are independent with marginal distributions $X_i$ ~ Poisson($\lambda(t_i - t_{i-1})...
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Finding the conditional distribution of a poisson process

This question is from a workbook i'm currently working on. If we have a poisson process thats is on a real line and denote it with $S(x_1,x_2)$ as the number of events in the time interval between $...
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What's the probability that B will catch 3 fish before A catches 3 fish

This is a basic question, but I do not completely understand. A and B are both catching fish at times of independent poisson processes with rates $1$ and $2$ respectively. What is the probability B ...
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Intuition behind a conditional expectation in a Poisson process

Earthquakes occur according to a Poisson process with $\lambda = 5$ per year, let $T_1$ be the time of the first occurrence of the process and we wish to figure out what is the average time that will ...
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Campbell Theorem for $n$-th moment

Campbell's theorem gives a method for calculating expectations of sums of measurable functions f(x). While I was solving my system model considering a Poisson point process, I came across the ...
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Characterisation of a Poisson process

Consider a non homogeneous Poisson process, $N(t)$ with rate $r(t)$. The probability distribution of the first event $T_1$, conditioned on the fact that this happens in a finite time (not assuming ...
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Average response/waiting time for aggregated tasks with Poisson arrival

Suppose there is a specific computation task with Poisson arrival rate $\lambda$ that could be aggregated in a way that when a task arrives and triggers a computation which lasts for $D$ seconds, if ...
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Distribution of the maximum minus the minimum of 2 exponential RVs

Let $$S\sim Exponential(\lambda), T\sim Exponential(\mu), U=min\{S,T\}, V=max\{S,T\}$$ The question doesn't say whether $S$ and $T$ are independent but I'm assuming they are, because I sort of used ...
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Counting uniform RVs on [0,1] partitioned into k subsets

In this problem, we have a series of iid uniform rvs on [0,1] and call these RVs particles. We also partition [0,1] into k non overlapping subsets. We define $N$ to be a Poisson process that specifies ...
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Poisson process arrival time.

let $N_t$ be the Poisson process with parameter $\lambda$ and $S_{n}$ denote the time of the $n^{th}$ event. define $T_i:=S_i-S_{i-1}$ find $E(S_{4}|N(1)=2)$. on the one hand, use the method in ...
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modelling arrival events for systems with multiple users?

Assume that the system at hand has a set of users M = {M1, M2, ...., Mn} and each Mi has a probability of creating an event e.g., M1=0.3, M2= 0.1, where the total probability for all users =1. The ...
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Leaky integrate-and-fire neuron firing rate for poisson input

A leaky integrate-and-fire neuron is given by the following ODE $\dot{V} = -\frac{V}{\tau} + \dot{V}_{input}(t) - V_{max}S(t) $ The neuron produces a spike every time its membrane potential $V$ ...
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Is Poisson Process a time series of Poisson Distribution?

Is Poisson Process a time series of Poisson Distribution? If Yes, can anyone explain it using a movie rent example?