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.
Questions tagged [poisson-process]
1414 questions
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Why $\mathbb P\{N(t+\Delta t)-N(t)=1\}=0+\lambda\Delta t+o(\Delta t)$ if $N$ is a poisson process.
I have the following definition:
Definition (Poisson process). $N(t)$ is a poisson process if $N(t+h)-N(t)$ and $N(t)-N(t-k)$ are independent and $N(t+h)-N(t)\sim N(h)-N(0)$.
We try to find the law of such a process. In my course it's written that…
idm
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Motorway problem: how to minimize the number of encounters?
I was working on the motorway problem:
At time t=0, cars are launched from the same entrance of highway following a Poisson process with parameter $\lambda$, and the speed of cars ($v$) follows a known distribution of $S$.
The observer is on one of…
Heifetz_Fan
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Poisson process for expected customer arrival
Customers arrive at a service facility according to a Poisson process of rate $\lambda = 5$ customers/hour. Let N(t) be the number of customers that have arrived up to time t hours. Let $W_1,W_2,W_3,...$ be the successive arrival times of the…
waterr
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Density of $T_2 - T_1$ of a non homogeneous poisson process with $\lambda$(t) = $\frac{1}{1 + t}$ intensity
Me and my teammate have this question to solve in our homework. The question is the following:
A N(t) non homogeneous Poisson process with intensity $\lambda$(t) = $\frac{1}{1 + t}$. Find the density of $T_1$ as well as the density of $T_2 -…
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o(h) properties within a poisson processes
I have this as part of a theorem for the sum of two poisson processes being itself a poisson process. I don't particularly understand why the fourth line equals the last line of the theorem. I believe there should be a λµh^2 in the last line but…
loaf
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Unnormalized Arrival Times of Nonhomogeneous Poisson Process
Consider a nonhomogeneous Poisson process $N(t)$ for $t\ge0$ defined by the instantaneous intensity $\lambda(t)\ge0$ and mean value function $\Lambda(t)=\int_0^t\lambda(s)ds$. We know…
Andras Vanyolos
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Show the independence property for Standard Homogenous Poisson Process
Problem:
Let $\tilde{N}$ be a Standard Homogenous Poisson Proces on $[0,\infty)$, and let N be a Poisson process on $[0,\infty)$ with mean value function $\mu$
Show that $N=(\tilde{N}(\mu(t)))_{t\geq0}$ is a Poisson Process on $[0,\infty)$ with mean…
Flems
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IFOA CS2 cases when compound Poisson Process is also a Poisson Process
$S_{t} = X_{1} + X_{2} + ... + X_{Nt}$ is a compound Poisson process.
I know that when $X_{j}$ can only take value 0 or 1, $S_{t}$ is also Poisson process.
I can understand when $X_{j}$ = 1 for all j, $S_{t}$ is a Poisson Process. But why when some…
user196736
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Show the increment property for $N = N_1 + N_2 + ... N_n$
Problem:
Let $N_1 , ...., N_n$ be independent Poisson processes on $[0,\infty)$ defined on the same probability space.
Show that $N_1 + N_2 + ... + N_n$ is a Poisson process and determine its mean value function.
Attempt:
I have shown all other…
Flems
- 416
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Poisson Process Arrival Rate Estimation
Poisson Process Arrival Rate Estimation
I think we are supposed to use the fact that Poisson is memoryless but I don't understand how that helps us in calculating the arrival rate.
DiamondCorp
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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$(\lambda)$ random variables. Let $T_n=\tau_1+\dotsc+\tau_n$…
togashi13
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Poisson process in a restaurant
While preparing for a midterm, I came across this question
Suppose a restaurant is visited by 10 clients per hour on average, and clients follow a homogeneous Poisson Process. Independantly of other client, each client has a 20% chance to eat here…
Bozu
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Poisson Process -
Cafe shop opens at 8:00. Customers arrive at rate of 35 customers per hour. Find the probability that between 80 and 100 customers arrive between 10:00 and 13:00.
Just looking if I am correctly thinking.
I have modeled the question like this:…
vekeras
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Properties of the intensity (or rate) function of a Non-homogeneous Poisson process
A non-homogeneous Poisson Process is parameterized by its intensity (or rate) function $r(t)$ for $t \in [0, \infty)$. Often what is assumed in the literature about the function $r$ is that $$\int_0^{\infty}r(t)dt = \infty.$$ I understand that this…
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Hitting time of Poisson process is finite
let us consider Poisson process $N_t$ with $\lambda$ parameter and a stopping time $T=\inf\{t\ge 0;\,N_t=a\}$, where $a\in\mathbb{N}$. I would like to show that $ET=\frac{a}{\lambda}$, so I want to use Doob Theorem. I want to show that…
mwr
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