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]
1349 questions
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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
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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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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
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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
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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
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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
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Memoryless property for Poisson process
Suppose that arrivals of a certain Poisson process occur once every $4$ seconds on average. Given that there are no arrivals during the first $10$ seconds, what is the probability that there will be 6 arrivals during the subsequent $10$…
user3727610
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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…
Michael Hardy
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Strong law of large numbers for Poisson process
My question regards the strong law of large numbers as stated, e.g., in Ethier and Kurtz (1986, p. 456 Eq. (2.5)), as follows:
If $Y$ is a unit Poisson process, then for each $u_0>0$,
\begin{eqnarray*}
\lim_{n \to \infty} \sup_{u \geq u_0} \vert…
Mareen
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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
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Applications of the Mecke formula
The Mecke formula as defined in the book Lectures on the Poisson Process (page 27) is:
The text states (bottom of page 26) "This equation is a fundamental tool for analysing the Poisson process and can be used in many specific calculations." I can…
Olivier Bégassat
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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
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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
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Independent increments in the Poisson process.
Let $(N_t)_{t\geq0}$ be a Poisson process. For me, the definition is the following: $N_t=\max\{n\geq0:T_n\leq t\}$, where $T_0=0$ and $T_n=S_1+\ldots+S_n$, where $S_1,\ldots,S_n$ are independent exponential random variables of parameter $\lambda$…
user39756
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How to show that $(X(t)-\lambda t)^2 - \lambda t$ is a martingale, where $X(t)$ is a Poisson Process?
I am trying to show that $(X(t)-\lambda t)^2 - \lambda t$ is a martingale, where $X(t)$ is a Poisson process with rate $\lambda$. So far, what I have done is:
\begin{align*}
E\left((X(t)-\lambda t)^2 - \lambda t| \mathcal{F}_s\right)&=…
user136503
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