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]
1457 questions
13
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What is the probability of getting an even number from a Poisson random draw?
Below is a graph showing the probability of drawing an odd number (y-axis) from a Poisson distribution with a given expected value (x-axis)
x = seq(0,1e4,1) // range of values to explore
lambdas = seq(0,4,0.01) // expected value of the…
Remi.b
- 1,655
12
votes
1 answer
Proving properties for the Poisson-process.
Define a Poisson process as a Levy process where the increments have a Poisson distribution with parameter $\lambda$*"length of increment".
I want to prove these properties:
It has almost surely jumps of value 1.
It is almost surely increasing.
When…
user119615
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11
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1 answer
A question about Poisson processes
Reading through the book "Brownian Motion & Stochastic Calculus" by Karatzas and Shreve,
I found the following exercise (problem 3.9, page 15):
Let $ \ N \ $ be a Poisson process with intensity $ \lambda > 0 $ (this means, in particular, that $ N_t…
Mad Si
- 893
10
votes
2 answers
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 customer can only order one coffee at a time and…
konofoso
- 681
10
votes
1 answer
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 our country are randomly located (independently…
Dan
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10
votes
2 answers
Splitting Poisson process formal proof
Let $\{X_t\}_{t\ge 0}$ be a Poisson Process with parameter $\lambda$. Suppose that each event is type 1 with probability $\alpha$ and type 2 with probability $1-\alpha$. Let $\{X^{(1)}_t\}_{t\ge 0}$ the number of type 1 events up until time $t$ and…
user128422
- 3,087
10
votes
1 answer
Concerning an infinite server queue with Poisson arrivals
Here's the statement of the problem (from Ross's Introduction to Probability Models):
For those unfamiliar with "infinite server queues," they are described here. In this case, however, the service times are not exponentially distributed; rather,…
thisisourconcerndude
- 3,281
9
votes
1 answer
Interarrival times of nonhomogeneous Poisson process
It is well known that interarrival times of homogeneous Poisson process are independent and exponentially distributed.
But how about interarrival times of nonhomogeneous Poisson process:
- are they still independent random variables?
- what is…
tern
- 123
8
votes
1 answer
Average queue length with impatient customers
Suppose customers join a queue with a poisson arrival rate $m$. If a customer is not served within a unit of time, she abandons the queue. Customers are served in a first-come-first-served (FCFS) manner. There is a single server, and the service…
afshi7n
- 267
8
votes
2 answers
Distribution of interarrival times in a Poisson process
I am new to Statistics. I am studying Poisson process, I have certain questions to ask.
A process of arrival times in continuous time is called a Poisson process of rate $\lambda$ if the following two conditions hold:
The number of arrivals in an…
Supreeth Narasimhaswamy
- 1,272
8
votes
0 answers
Average shape of Voronoi cells in dimension $n\ge 2$?
A Poisson point proess of constant intensity in $\mathbb R^n$ has a Voronoi diagram. It is known that when $n=2$ the average number of edges of a cell is exactly $6$. Last I heard (but that was a while ago), the probability distribution of the…
7
votes
3 answers
Poisson distribution and the choice of unit
Suppose that the average number of cars passing by a building, $X$, is 3 cars per minute. Assuming that $X$ follows the Poisson distribution with mean $\lambda=3$, the probability that 1 car passes per minute…
ashpool
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7
votes
2 answers
Expectation and martingale properties of integral of Poisson process w.r.t. itself
Let $N_t$ be a Poisson process with rate $\lambda$ and let $M_t=N_t-\lambda t$.
I am then trying to find
$$
\mathbb{E}\left[\int_0^tN_s\,\mathrm{d}M_s\right].
$$
I have tried applying the definition of an Îto integral to find…
Renze
- 119
7
votes
0 answers
Approximation of Poisson by Wiener
Recently I learned that it is a widespread idea in applied math to approximate high rate Poisson processes by a Wiener process. I.e. take $N$ to be a homogeneous Poisson with rate $\lambda$, then for a large enough $\lambda$ one can select some time…
demitau
- 837
7
votes
3 answers
Let $U=\operatorname{min}\{X,Y\}$ and $V=\operatorname{max}\{X,Y\}$. Show that $V-U$ is independent of $U$.
Let $X$ and $Y$ be exponentially distributed random variables with
parameter $1$ and let $U=\operatorname{min}\{X,Y\}$ and
$V=\operatorname{max}\{X,Y\}$. Show that $V-U$ is independent of $U$.
We have shown that $U$ is distributed…
Aka_aka_aka_ak
- 1,649