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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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What is “concretely” $\int_a^b f(t)dN_t$ when $N_t$ is a Poisson process?

What is "concretely" $\int_0^1 f(t)dN_t$ when $N_t$ is a Poisson process ? In the sense, what is the interpretation ? Is it something as $$\lim_{n\to \infty }\sum_{k=0}^{n-1}f(t_i)(N_{t_{i+1}}-N_{t_i})...
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Martingales related to the square of a compensated Poisson process. [duplicate]

Let $N_t$ be a homogeneous Poisson process with rate $\alpha>0$ and $M_t=N_t-\alpha t$. Show that $M_t^2-\alpha t$ and $M_t^2-N_t$ are martingales. I computed \begin{align} \mathbb E[M_t^2\mid \...
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Is there a sort of Poisson process s.t. many people can arrive in the same time?

Suppose that a bus has 30 places. A way to model in how long the bus is full where people come independently is to use Poisson process. But in this type of model, we only consider people that come "...
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Difference between Poisson processes and Poisson distribution

We suppose that a factory has on average 3 call per minutes. What is the probability to have 3 call in 4 minutes? I'm always confuse. Should I use a Poisson random variable or a stochastic process? i....
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Especifying a measurable space for a homogeneos Poisson process.

I'm studying about the Poisson process (PP), and so far I can not find anything about the measurable space that the PP is defined. Then, I would like to know what is the measurable space for a ...
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Bulbs with amnesia

Here is a question for which I am not able to figure out the approach to solving it. Problem statement: Suppose that $n$ light bulbs in a room are switched on at the same instant. The life time of ...
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Unbiased estimation of Poisson process

We have a sample $X_i$, where $i=1...n$ and $X_i \sim Poi(\lambda)$. The problem is to find a constant so that $\frac{1}{2} s^2 + cM(X)$, where $s$ is the corrigated sample standard deviation and $M(X)...
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Alternate characterization of a spatial Poisson point process

Would I be correct to assess that a spatial Poisson point process on some compact, say the $d$-dimensional sphere, can be simulated by first choosing some $n \sim \mathrm{Poisson} (\beta)$ number of ...
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Probability of uniform distribution with poisson process

Men and women arrive at a store according to independent Poisson processes with hourly rates $\lambda_M$ = 3 and $\lambda_F$ = 4, respectively. Men shop for a time that is uniformly distributed on [0, ...
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Distribution of the residual lifetime for a Poisson process.

Let $\{N(t):t\geqslant 0\}$ be a Poisson process with intensity $\lambda>0$, interarrival times $T_n$, and $S_0=0$, $S_n=\sum_{i=1}^n T_i$ for $n\geqslant 1$. Define the residual lifetime process ...
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Renewal reward process

Consider a train station to which customers arrive in accordance with a Poisson process having rate λ. A train is summoned whenever there are N customers waiting in the station, but it takes K units ...
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Some limit problem in a light-tailed Compound Poisson driven queue

As part of a homework exercise on a light-tailed Compound Poisson driven queue (i.e. a Compound Poisson process reflected at 0) I need to show that when $B$ denotes the job size of the Compound ...
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Poisson process on Skorokhod's space

For each $n=1,2,\ldots $, let $\ \xi_{n1},\ldots, \xi_{nn}$ be random and independent variables such that $\mathbb{P}(\xi_{ni}=1)=p_n \ \ $ and $\ \ \mathbb{P}(\xi_{ni}=0)=1-p_n$. Let consider the ...
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Poissonian nature of photon count

I am trying to use poissonian distribution to validate photon emission of x-ray source. Photons counts are recorded at 100ms intervals using a photon counting detector. If the photon distribution is ...
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Prove for compound Poisson process $\mathbb{E}e^{i\theta X_t}=\mathrm{exp}(-\lambda t\int (1-e^{i\theta x})\mu(dx)),\ \theta\in\mathbb R$.

Let $N=(N_t)_{t\ge 0}$ be a Poisson process with parameter $\lambda$, and $(\xi,\xi_1,\xi_2,\cdots)$ be an independent sequence of i.i.d. random variables that are independent of $N$, such that the ...
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A company can employ 3 workers, what is the company’s long-run average daily revenue?

A company can employ 3 workers. Each worker independently stays on the job for an exponentially distributed time with a mean of one year and then quits. When a worker quits, it takes the company an ...
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Find probability it takes less time to send letters according to a Poisson process with intensity $1$, then send them one by one with intensity $5$.

At the center of espionage in Kznatropsk one is thinking of a new method for sending Morse telegrams. Instead of using the traditional method, that is, to send letters in groups of 5 according to a ...
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Poisson process with fixed number of events

There is a question in a textbook which states the following: Arrivals are modelled as being from a Poisson process with rate r per minute. Given that 2 arrivals occur in a particular minute, at what ...
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How does the solution $U∗z(t)$ work for renewal equation?

This post discusses Example 3.5.4 in Resnick's book on page 203. However, when I try to understand the next example 3.5.5, I still have some doubts. Example 3.5.5 If $F(dx) = xe^{-x}dx$, then we ...
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Proving a that the arrival times of a Poisson process is uniformly distributed

I am stuck on this question for a very long time and I can't figure out the solution: "Suppose $N_t$, $t\ge0$, is a Poisson process with rate $\lambda$ and $N_t = 0$. $T_0$ and $T_1$ are the first ...
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Is a Poisson r.v.'s parameter a rate $\mu$ or a count $\mu t$?

Let's say I want to model the arrivals of some quantity of interest, say customers coming to a store. I know that on average, $\mu$ customers arrive in on hour. My understanding is that if $N$ is the ...
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Laplace transform gives a wrong result when finding Renewal function

In page 188 of the book Adventures in Stochastic Processes, it shows that if $F(dx) = xe^{-x}dx$, then the renewal function $U(x)$ will have the following expression $$U(x) = \frac{3}{4}+\frac{x}{2}+\...
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Poisson Process example from Durret's Probability textbook

I'm struggling to understand some examples related to the following theorem in Durret's Probability: Theory and Examples. The theore is: For each $n$, let $X_{n,m}, 1 \leq m \leq n$ be independent ...
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Closed form solution for the difference of two poisson processes

I'm interested in whether there is a closed-form distribution of the time it takes two Poisson processes to output counts to have a fixed difference. For example, let $k_1$ ~ Poisson($\lambda_1t$) $...
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Explicit formula for the one-dimensional distributions of a time-homogeneous Markov chain subordinated by a Poisson process

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(E,\mathcal E)$ be a measurable space, $(X_n)_{n\in\mathbb N_0}$ be a $(E,\mathcal E)$-valued time-homogeneous Markov chain on $(\...
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Is this Poisson Process problem worked out correctly?

Calls are received at a company call center according to a Poisson process at the rate of five calls per minute. (a) Find the probability that no call occurs over a 30-second period. (b) Find the ...
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Are these Poisson-related problems and are the solutions correct?

In a city there are three kinds of subway lines: the red, green and orange lines. Subways on each line arrive at a station according to three independent Poisson processes. On average, there is one ...
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Expectation of arrival times.

Let $(N_t)_t$ be a Poisson process with parameter λ = 2. By $τ_k$ denote the time of the k-th arrival (k = 1, 2, . . .), and by $ρ_k = τ_k −τ_{k−1}$ - the interarrival time between the (k−1)th and kth ...
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What is the difference between time, arrival-time, and inter-arrival-time is Poisson process?

Let $(N_t)_t$ be a Poisson process with parameter λ = 2. By $τ_k$ denote the time of the k-th arrival (k = 1, 2, . . .), and by $ρ_k = τ_k −τ_{k−1}$ - the interarrival time between the (k−1)th and kth ...
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Why can't I write $P(X>5|X>1) = P(X>5)$?

I have a confusion with the memorylessness property of Exponential distribution. If exponential distribution is memoryless (i.e. the past has no bearing on its future behavior), why can't I write $P(...
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How are $P(N_1=2, N_4-N_1=4)$ and $P(N_1=2, N_3=4)$ equivalent in a Poisson process?

$$P(N_1=2, N_4=6) = P(N_1=2, N_4-N_1=4) = P(N_1=2, N_3=4) = P(N_1=2) \cdot P(N_3=4)$$ What I understand is: $P(N_1=2, N_4=6)$ means "the probability of count of 2 items arriving at step-1, AND 6 ...
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How does the equation make sense in case of a Poisson process?

$$P(N_1=2, N_4=6) = P(N_1=2, N_4-N_1=4) = P(N_1=2) \cdot P(N_3=4)$$ How does the above equation make sense in case of a Poisson process? Can anyone explain?
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Find the expectation of the time difference between two consecutive events when the total number of events is given.

Suppose we have a Poisson process with $\lambda$, I am trying to find the expectation of the time difference $\Delta_t$, i.e., $\mathbb{E}(\Delta_t)$, between two consecutive events when the total ...
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Calculation of expectation of Poisson Process.

This is a problem related to Poisson Process where $\lambda = 2$. $ E(N_3N_4) \\ = E[N_3(N_4-N_3 + N_3)] \\ = E[N_3(N_4-N_3) + N^2_3)] \\ = E[N_3 - N_0(N_4-N_3) + N^2_3)] \\ = E[N_3 - N_0(N_4-...
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Are the two random variables independent?

Consider the following two random variables, In first case you record the number of people arriving at a queue, for a random amount of time. Note, here the arrival of people in the queue is random (...
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Probability Problem: Combination of Poisson and Binomial

A hitchhiking gentleman waits by the roadside for cars to pass by that will take him to his destination. The daily amount of cars that pass by him that are going to his destination behaves in the ...
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Computing the conditional probability of a Poisson process with thinning.

Let $\{N(t) \}_{t\geq0}$ be a Poisson process with rate $\lambda$. Suppose that each "arrival" is independently of type "$i$" with probability $p_i$. Then I know that the $\{N_i(t) \}$ are independent ...
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Poisson process' waiting time

Studying homogeneous Poisson process I have the next problems: Passengers arrive to a bus station according to a homogeneous Poisson process with $\lambda = 4$ per minute. If the bus waits for $20$ ...
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Probability of falling meteor (Poisson process)

Let's say, that we know, that in some period of year there are really common sightings of meteors with an average of 100 meteors per hour. What's the chance of no meteor seen in 5 minutes? I was ...
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Problem about non-homogeneous poisson process

$\textbf{Problem}$ Suppose customers arrive at a system according to the poisson process with rate $\lambda$. Every customer stays in the system for an exp($\mu$) amount of time and then leaves. ...
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If an integral is P-measurable with respect to a G measure, then this integral will be P-measurable with respect to a PN measure, since $PN \ll G$?

I am struggling to prove a result, which I don't know if it is possible as I have not found any reference. I will write some information that I don't know if they are really necessary to solve my ...
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Modelling Sales in a software company: Poisson or simple linear algebra?

Let's assume I have 1 year of weekly sale data for software A and 1 year of weekly sale data for software B. Software B is related to software A, because it's a maintenance/security upgrade, so the ...
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Find conditional probability of given Poisson process

A Poisson process have rates $\lambda$, and given $0<s<t$. I want to find $P(X(t)=n+k\vert X(s)=n)$. I have tried like this \begin{eqnarray} P(X(t)=n+k\vert X(s)=n) &=& \dfrac{P(X(t)=n+...
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Row-Sum for a Poisson Process

We know that for a Markov process with discrete state space, the sum of the transition probability function over its state space should be 1. For a Poisson process that is, 1 + o(h). Could anyone ...
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Poisson Process from independent non-identical exponential RVs

I know, I can define a Poisson Process using a sequence of i.i.d. exponential random variables, i.e. let $\tau_1, \tau_2, \tau_3, ... \sim \mathrm{Exp}(\lambda)$, then $T_i = \sum_{j=1}^I \tau_j$ are ...
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Is sum of two sequences of independent i.i.d random variables independent?

I am looking for a proof that if I have two sequences of i.i.d random variable that are independent $$ (X)^\infty_{n=1} , (Y)^{\infty}_{n=1} $$ then its' sum $ (X+Y)_{k} $ is also independent from $...
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What's the probability that zero meteors were seen in the first hour but at least 10 were seen in the last three hours? (Poisson process)

So the setup of the question is that a group of students is watching a meteor shower from $11$ p.m. to $3$ a.m. and they arrive according to a Poisson process with intensity $\lambda = 4$. The ...
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Combination of Exponential distributions question (with different probabilities)?

I am currently learning about Poisson processes and I was thinking about the following question. The waiting time for your lunch to be collected by delivery drivers A and B are distributed as $\...
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Joint Distribution for Number of Arrivals

Suppose that the number of women who buy concert tickets follows a Poisson process with rate $30$ women per hour, and similarly the number of men who buy tickets is a Poisson distribution with rate $...
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Let $\{X_t ,t\ge0\} $ be Poisson process with parameter $\lambda $ and arrival time $T_1,T_2,\cdots$ , Find $Var(T_2-t|X_t=1)$

Let $\{X_t ,t\ge0\} $ be Poisson process with parameter $\lambda $ and arrival time $T_1,T_2,\cdots$ , Find $Var(T_2-t|X_t=1)$ My try: First I want to find the conditional density of $T_2-t$ given $...