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.
1,456 questions
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Lengths of Segments Formed by Binomial Point Process
Suppose you have a Poisson Point Process $\{X_k\}$ with intensity $1$ on $\mathbb{R}$, and we truncate it on $[0,L]$. For a positive integer $n$, I would like to compute or estimate
\begin{equation*}
\...
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Proof of Poisson process property with random time
I have written the following statement and its proof.
Statement:
Let $\{N_t, t \geq 0\}$ be a Poisson process with rate $\lambda$ and let $T$ be a random variable independent of $N_t$ and $N_{t+h} - ...
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Expected waiting times for buses with Poisson arrivals
I have several related questions about waiting for buses that follow Poisson processes.
Setup: Two bus stops near my home. Bus A arrives every 12 minutes on average (rate λ_A = 1/12), Bus B every 15 ...
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Finding a closed form for a Maclaurin series summation in a compound Poisson process
I was playing with a compound Poisson process for rainfall where the $X_1,X_2,...,X_N$ are exponentially distributed and $N(t)$ is of course a Poisson random variable. While finding the marginal pmf ...
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State-dependent Poisson process
In general, the analysis of a homogenous or non-homogenous Poisson process $X(t)$ is well known (we can compute waiting time, master equations, etc.). Here denote the rate as $\lambda$ (or $\lambda(t)...
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Construction of Poisson Process via arrival times
I am trying to prove a well known fact that there is an alternative characterisation of Poisson process using renewal times, as claimed in the following paragraph:
I am interested in proving ...
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Doubt in Possion and Exponential Distribution co-relation
I was solving the following problem from debore:
At time t = 0, 20 identical components are tested. The lifetime distribution of
each is exponential with parameter λ. The experimenter then leaves the ...
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How to derive the joint distribution of $T_1, T_2$ given $T_3$ in Poisson Process
Let $T_n$ be the time until the $n$th arrival in a Poisson Process. A problem I am trying to solve asks for the joint distribution of $T_1$ and $T_2$ given that $T_3 = s$ for some $s > 0$. Let $S_i$...
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A doubt from Durrett's book on probability theory on construction of a Poisson Process
I am recently studying stochastic processes and I encountered the following theorem in Durrett's book Probability: Theory and Examples (fourth edition). In page 155-156, he considers $X_1,X_2,....$ ...
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Autocorrelation function for a Compound Poisson Process?
Let $N(t)$ be the number of events that occurred up to time $t$ for a Poisson process.
It is relatively easy to compute the autocorrelation function for this process, which is:
\begin{equation}
E[N(...
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Can we estimate an inhomogeneous Poisson process with a Gaussian prior from data?
Suppose I have a an inhomogeneous Poisson process and some data $0< t_1 < t_2 < ... < t_n < T$.
$\{t_1, t_2, ..., t_n\} \sim \mathrm{PoissonProcess}(\lambda(t))$
Now I further specify ...
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What is the distribution of order in ordered sets of poisson processes
I have a question from epidemiology that I'm struggling 1) to write down mathematically and 2) determine if it has a closed form solution.
First here's the epidemiological question. Say I have $C$ ...
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how do I quantify likelihood of bad data - Bayes / Posterior distribution
Problem: I have a ratio statistic and I want to separate out bad data and flag accounts that have unreliable data for a particular feature of a predictive model.
The ratio is # events per 1000 mile
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Martingale with Poisson Process
I came across the following exercise: Let $N$ be a Poisson process with intensity $\lambda$ and $\{Y_k\}_{k=1}^{\infty}$ a sequence of positive i.i.d random variables satisfying $\mathbb{E}[Y_1] < \...
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Probability of that a compound Poisson process is continuous on an interval $[0,N]$
Let $N=(N_t)_{t \geq 0}$ be a rate $\lambda$ Poisson process and let $(Y_k)_{k\geq 1}$ be iid random variables independent of $N$. Define the compound Poisson process $Z=(Z_t)_{0\leq t \leq 1}$ as ...
user1253808
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Computing the expected minimum of two sums of two independent Exponential variables
Given some independent and identically-distributed random variables $X_1, X_2, Y_1, Y_2 \sim \text{Exp}(\mu)$, I'm trying to compute
$$
E[\min{(X_1 + X_2, Y_1 + Y_2)}]
$$
in order to model the ...
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Convergence speed of decreasing sequence of points in a Poisson Random Field
I am stuck at one exercise about the Poisson Random Field I have trying to solve for the last days.
We are in the standard probability setting, given is $\beta \in \mathbb{R}_{> 0}$ and a Poisson ...
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Poisson process without the indepedent increments?
From Wikipedia:
A counting process is a homogeneous Poisson counting process with rate $\lambda > 0$ if it has the following three properties:
$N ( 0 ) = 0$;
has independent increments;
the number ...
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A Poisson Process in which the rate of events increases as more event happens?
I am trying to define a (Non-Homogenous) Poisson Process such that the rate of events increases as the cumulative number of events increases.
I tried to do this as follows:
First, I defined a rate ...
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Is it better to hire $1$ fast barista vs $2$ slower baristas?
This is another math puzzle I heard today.
Consider a M/M/K queue (https://en.wikipedia.org/wiki/M/M/c_queue) in a cafe. Lets say the cafe has a rule that each queue is FIFO (first in first out), each ...
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Time needed to observe $k$ events in a Poisson Process?
Consider a Poisson process.
Let $N(t)$ be the number of events that have occurred up to time t. Then, for any time interval $(t, t+h]$, the probability of k events occurring in this interval is given ...
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Approximating a Communication Traffic Model with a Poisson Process
I’m working on a problem related to communication networks, specifically modeling traffic at a network node. The traffic arrives at a relatively high rate, and the model I’m dealing with is not in a ...
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density of the vector of jump times in a Poisson process
Let $(N_t)$ be a Poisson process. We let
$$
T_k=\inf \lbrace t\in \mathbb{R}: N_t\geq k \rbrace.
$$
Let $n\in \mathbb{N}$ and $t\in\mathbb{R}^+$. I want to prove that the conditional density of the ...
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How can I calculate the rate of overlapping events between two Poisson processes?
I have the following problem: I have 2 independent Poisson processes $N_1$ and $N_2$ with known rates $\lambda_1$ and $\lambda_2$ respectively.
The events in each process have a well-defined duration, ...
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Equivalent characterizations of a Poisson process
Consider a counting process $N(t)$, and the following three conditions that I believe are equivalent:
$$
\begin{aligned}
(A) \quad& N(t) \text{ is a Poisson process of intensity }λ \\
(B) \quad&...
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