# 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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• 103
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### Probability 2 earthquakes happen in a period of time.

The amount of earthquakes that happen at island X follows the Poisson process with mean 2 . Given that 2 earthquakes have happened in this year, find the probability both the earthquakes happen ...
• 135
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### Interarrival Times for a Non-Homogeneous Poisson Process

It is well known that the interarrival times for a standard (i.e. homogeneous) Poisson Process follow an Exponential Distribution (What is the correct inter-arrival time distribution in a Poisson ...
• 685
1 vote
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### Relationship Between Poisson Process and Birth and Death Process

I am learning about Birth and Death (Stochastic) Processes (https://en.wikipedia.org/wiki/Birth%E2%80%93death_process): When a birth occurs, the process goes from state $n$ to $n + 1$. When a death ...
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### Probability distribution function that governs the number of points in histogram bins. Consistency between multiple dimensions?

Let's have a continuous random variable $x$ with some probability density function $f(x)$. A bin is given by an interval $[x_1, x_2)$. Given a total number of points $N$ generated from the PDF $f(x)$, ...
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1 vote
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### When can we compute the CDF of the Compound Poisson process

Suppose we have a Compound Poisson Process with intensity $\lambda$ $$S_t = \sum_{i=1}^{N_t} X_i.$$ We can compute the formula for the CDF as follows \begin{align*} F_{S_t}(x) = P(...
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### M/M/c Queue Model Solutions for Average Waiting Time and Queue Length

I am seeking assistance with a queueing theory problem involving the M/M/c queue model from my textbook. I have attempted to solve the problem and would greatly appreciate it if someone could review ...
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### Questions about Probability: Theory and Examples problem 5.3.8: why do we have $\lambda<x(1-p)$

I was doing Probability: Theory and Examples problem 5.3.8 To use Theorem 5.3.8, we need to prove $\mathbb{E}_x[\phi(X_1)]<\phi(x)$, which is just $\mathbb{E}_x[X_1]<x$. After some simple ...
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### Expected number of passengers in a bus, if both bus and passengers arrival time have a Poisson distribution.

Here's the full question In Poisson Bus City, there is a shuttle bus that goes between Stop A and Stop B, with no stops in between. The times at which the bus arrives at Stop A are a Poisson point ...
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### expected value of exponential compound poisson process

Let $Z(t)=\sum_{i=1}^{N(t)} X_i$ and let $N(t)$ be a Poisson process with parameter $\lambda$ and $X_1,X_2,\dots$ positive iid random variables with density function $f_X(x)$, independent of $N(t)$. ...
• 91
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### Customers arrive at a facility according to a Poisson process

Customers arrive at a facility according to a Poisson process $N(t)$ of rate $\lambda = 5.5$ customers/hour. Each customer is admitted to the facility with probability $p=0.6$. All customers, who are ...
• 71
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### Characterization of compensated Poisson processes

I've found the following statement (but it is rather an example) in a book that states if $X$ is a local martingale and its quadratic variation has the form $$\left[X\right]=t+cX$$ where $c$ is a ...
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• 1,479
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### How to calculate the expectation of this Poisson-like process?

Question: Let $\tau_i\sim \text{Exp}(\lambda_0)$ iid and $\gamma_i\sim \text{Exp}(\lambda_1)$ iid and independent of each other and set $N_t=max\{k\geq 0: \sum_{i=1}^k{(\tau_i+\gamma_i)}\leq t\}$. ...
1 vote
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### What rule is used in this derivation of the interarrival time for the Poisson process?

I'm working on calculating the probability distribution of the interarrival time of the Poisson process. The method used in my textbook is very strange I don't understand how the probabilities are ...
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### Regarding Construction of a Poisson Process

I'm taking a graduate level course on Stochastic Processes and encountered the following problem in one of our assignments. $\textbf{Problem:}$ Fill in the details of the of the following construction ...
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• 71
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### Inductive proof Poisson process counts follow Poisson distribution

In converting between two definitions of a Poisson process, namely starting from the "exponential inter arrival-times" definition and attempting to prove the "Poisson distribution of ...
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### True or false: In a 2D Poisson process, for every point $P$, there exists a convex $1000$-gon with Poisson points as vertices, that contains only $P$.

I made a Desmos graph that generates $30$ uniformly random black points in a disk, with the centre of the disk in red. I asked myself, "Can I always draw a convex quadrilateral with four of the ...
• 25.7k
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### A walk on a $2D$ Poisson process in which every step goes to the nearest unvisited point: expected distance from origin after $365$ steps?

Uncle's epic journey One year ago, my uncle set off from our village on an epic journey, in which every day he travels to the nearest unvisited village and stays there for the night. The villages in ...
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### Higher conditional intensity implies higher variance of the resulting counting process

Suppose $N=(N_t)_{t\geq0}$ is a simple counting process that is driven by the conditional intensity process $(\lambda_t)_{t\geq0}$. That is, for $(\mathcal F_t)_{t\geq0}$ the natural filtration ...
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### Superposition of Poisson Processes with a weighted sum

If L(t) and M(t) are independent Poisson Processes, for which a and b is N(t) = aL(t) + bM(t) a Poisson Process. A necessary condition for it to be Poisson is if N(t) has Poisson distribution and a ...
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### What is the steady state distribution of this Poisson process with non-constant rate?

I am looking for the steady state distribution of the following Poisson process: $$d x(t) = -k_1(x(t)-k_2)dt + k_3dN(t)$$ where $k_1$, $k_2$ and $k_3$ are constants and the rate $\lambda(x)$ of the ...
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### A square contains many random points. From each point, a disc grows until it hits the nearest neighboring point. What is the total area of the discs?

A unit square lamina contains $n$ independent uniformly random points. Each point is the centre of a disc whose perimeter touches the nearest neighboring point. Here is an example with $n=20$. In ...
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### Probability Density Function of Compound Poisson Process

I am trying to determine if it is possible to compute the probability density function (PDF) of a compound Poisson process $Y(t) = \sum_{i=0}^{N(t)} X_i$, where $N(t)$ is governed by a Poisson process ...
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