# 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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### Statistical hypothesis [closed]

A die is tossed $6400$ times. If a $1$, $3$ or a $5$ is realized at any toss, a failure ($F$) is recorded. If there were $3195$ failures, formulate the relevant statistical hypothesis and use it to ...
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### When is the quadratic covariation between two Poisson processes zero?

Let us consider a probability space $(\Omega,\mathscr{F},\mathbb{P})$ endowed with a filtration $\mathbb{F}$, and let $N$ and $M$ be two independent Poisson process with different parameter $\lambda$ ...
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### Intensity estimation of the marked process.

I observe a sample of the form $𝑆=[𝑆_1,𝑆_2,…,𝑆_𝑁]=[(𝑡_1,𝑥_1),(𝑡_2,𝑥_2),…,(𝑡_𝑁,𝑥_𝑁)]$ where each $𝑡_𝑖$ is an arrival time and 𝑥𝑖 is the amount of money spent by the 𝑖th customer. I ...
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### Wrong expected value of sum of Poisson process wait times [duplicate]

Assuming a Poisson process $N_t$ and denoting wait times $S_k$ (i.e. times until the $k$-th jump), I want to find the expected value of their sum: $$\mathrm{E}\left[\sum_{k=1}^{N_t} S_k\right]\;.$$ ...
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### Compound Poisson Process infinitely many jumps?

When I look at the following compound Poisson Process, where $(N_{t})_{t\geq0}$ is a Poisson Process with Parameter $\lambda$ and $\xi_{i}$ are $\mathbb{R}$ valued i.i.d random Variables independent ...
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### Why use infinity as upper bound for Campbell's theorem in this analysis (stochastic geometry)

I am reading a few papers on stochastic geometry analysis of wireless networks and when modeling interference effects at a reference point, the upper bound for Campbell's theorem is set to infinity. ...
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### Markov chain extracted from Poisson process

Let $(N_t)$ be a Poisson process. I'm trying to show that, extracting the integer times, I get a Markov process, namely setting $X_n=N_n$ ($n\in \mathbb{N}$), the family $(X_n)$ is a Markov process. ...
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### role of the independence assumption for Poisson point processes

A random set $\Pi$ is a Poisson point process with intensity measure $\Lambda$ (a locally finite measure), if For every bounded, measurable set $A$, the number of points in $\Pi \cap A$ is Poisson ...
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### An expression for $P(N_s=k|T_n=t)$ where {$N_t ; t\geq0$} is a Poisson process with parameter $\lambda$

Problem Let {$N_t ; t\geq0$} be a Poisson process with parameter $\lambda$ Solve for $P(N_s=k|T_n=t)$ where $T_n$ is the arrival time of the $n$-th event and $s$ and $k$ are positive integers My ...
1 vote
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### Blackwell's argument: Quadratic variation as an upper bound to an expected value in Kingman's book on stochastic processes

Let $S$ be a Borel subset of a complete measurable metric space, and $S^{*}:=S\times(0,\infty)$. There exists a countable family of subsets $B_{1},B_{2},...\subseteq S$ with the property that for any ...
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