# 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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### Taylor Expansion of a function with random variables

Setting Let $X_t \in \mathbb{Z}^{0+}$ be a random variable at time $t>0$. We have two counting processes $N(t)$ and $M(t)$ with: \begin{align*} & N(0) = M(0) = 0, \\ & dN_t \sim \text{...
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### $M_t = \frac{1}{\sqrt{T_1}} \mathbb{1} (T_1 \leq t) - 2 \sqrt{T_1 \wedge t}, t\geq 0$ is a martingale

I try to solve an old exam question, but I find it difficult. Maybe someone can suggest a hint. Let $\{ T_i | i\in \mathbb{N} \}\subseteq \mathbb{R} _{\geq 0}$ be a homogeneous Poission point process ...
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### Probability of forming an n-gon [closed]

We have more than 3 lengths. Each length is exponentially distributed with parameter $\lambda$ and each is independent and identically distributed. I want to calculate the probability of constructing ...
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### Conditional expected value of Poisson processes

Find Find $E \{ \frac{ \xi(5) }{ \xi(7)+1 } \ | \xi(7) \}$ where $\xi(t)$ is standard Poisson process I can easily solve similar problems, where a random process is represented by an ordinary ...
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### A question on a compound Poisson process: $P(|\xi(t)|\leqslant0.3)\underset{t\rightarrow\infty}{\rightarrow}0$?

Let $\xi(t)$ be a compound Poisson point process such that the number of summands is a Poisson point process with parameter $\lambda$ and the summands are random variables: $P(\xi_k=\pm1)=0.5.$ Is it ...
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### Confusion on when to use CDF and Poisson process

I'm going through the MIT OCW probability course (6.041sc), but I'm having trouble on when to use CDF and the Poisson process. Here's the problem (Recitation 15, problem 1). Problem Statement: ...
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### Can't understand the proof of the Time-Rescaling theorem.

I was reading the following paper: The time-rescaling theorem and its application to neural spike train data analysis and I have some difficulties understanding the proof of the time-rescaling-theorem....
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### If $X$ is a Poisson process, what is $\mathbb{P}(X(B_{a}(x)) = 0 \text{ for some }x \text{ with } |x|=b)$ for $0<a<b$?

Let X be a Poisson process with intensity $1$ on $\mathbb{R}^2$. For $0<a<b$, what is the value of $$\mathbb{P}(X(B_{a}(x)) = 0 \text{ for some }x \text{ with } |x|=b)$$ where $|\cdot|$ is the ...
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### Why is a completely random measure mixing?

Let $S_x$ denote an operator on $\mathcal{M}_{\chi}^{\#}$ by $S_x\xi(\cdot)=\xi(\cdot+x)$, where $\xi$ is a random measure. Here, $\mathcal{M}_{\chi}^{\#}$ refers to the space of boundedly finite ...
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### Poisson processes of two airline companies. (two independent Poisson processes)

Easyjet and KLM planes request landing permission at Heathrow airport according to independent Poisson processes with intensities $\lambda$ and $\mu$ per hour, respectively for Easyjet and ...
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### Correlation of Markov process

I have a problem where cars are entering an area according to a homogeneous Poisson process, with a rate of $\lambda = 9$ cars per hour. 20% of the cars entering the area are of type 1, and 80% of the ...
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### A customer queue where service time is exponentially distributed and arrival governed by a Poisson process.

I have just started Poisson processes in my course on stochastic processes. We have just covered random sums and now I got the following tutorial/ exercise class question, but I don't quite see what ...
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### Generation of a Poisson process

I am having trouble understanding the Poisson process generated as the following (in MATLAB): Choose the number of points, i.e., $N$ and a parameter $\mu$; Compute $y = rand(N,1)$, i.e., $N$ random ...
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### counting processes, stationary and independent increments

I know that Poisson process has stationary and independent increments. Are there any other known discrete processes with these two properties?
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### Intuition Behind Independence of Thinned Poisson Processes

Let $(N_t)_{t \geq 0}$ be a rate-$\lambda$ Poisson process, and let $(X_i)_{i \geq 0}$ be IID $\text{Bernoulli}(1/2)$. Denote by $(N_t^i)_{t \geq 0, i \in \{0, 1\}}$ the thinned Poission processes ...
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### Need help validating a proof that for any point process with MTBF $t$, the events in an interval sized $u$ will be $\frac{u}{t}$

I started a bounty on this question here: General point process - expected number of arrivals within an interval. The premise is that we have a point process where the time between successive events ...
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### Statistics - Binomial and Poisson Distribution Problem

I am given: I get 11 text messages per hour according to a Poisson process. The probability that a given text message is from my mother is $0.62$. I then have to find the probability that I receive ...
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### Change of jump sizes in Poisson processes

After applying Ito's lemma, I arrived to the following stochastic differential equation: $$dg(Y_t)=\left[ g(Y_t+t)-g(Y_t)\right]dN_t + g^\prime (Y_t)\beta^t \lambda dt$$ where $g$ is continuous, $N_t$...
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### Finite intensity of Lévy measure implies compound Poisson process

Suppose $X$ is a Lévy process with triplet $(b,\sigma^2,\nu)$ and finite intensity, so $\nu(\mathbb R)<\infty$. Why does it follow immediately that the jump part of $X$ can be described by a ...
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### Simultaneous Poisson Processes of Cabs and Passengers

Consider a airport exit with people leaving with a poisson process rate of 1 per minute. From the exit they can catch a cab, which arrives with a poisson process of 2 per minute. A person will wait no ...
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### Poisson Point Process Closest Point

Suppose we have a Poisson point process, with intensity $\lambda$. I want to prove that the expected distance from the origin and the closest Poisson point is $\frac{1}{2\sqrt\lambda}$ when in two ...