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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Statistical hypothesis [closed]

A die is tossed $6400$ times. If a $1$, $3$ or a $5$ is realized at any toss, a failure ($F$) is recorded. If there were $3195$ failures, formulate the relevant statistical hypothesis and use it to ...
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When is the quadratic covariation between two Poisson processes zero?

Let us consider a probability space $(\Omega,\mathscr{F},\mathbb{P})$ endowed with a filtration $\mathbb{F}$, and let $N$ and $M$ be two independent Poisson process with different parameter $\lambda$ ...
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Negative values in Poisson process

I'm trying to make some sense of the following definition: A collection of random variables $\{ N_t \}$$_{t \geq 0}$ is called a Poisson process with rate parameter $\lambda > 0$ if $N_0 = 0$ $N_{...
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Predict the probability of rooms being filled from a negative binomial process for arrival

This is somewhat of a riddle. I am trying to model the probability of filling up party rooms and simulation is not allowed. I know that the arrival of customers in 1 period of time (1 week) can be ...
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Expectation of waiting time ratios

Question: At a restaurant's bathroom, men and women arrive at the same rate, on average 20 minutes. Assume the number of people who waiting in the queue is Poisson distributed. Men spend on average 60 ...
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What is the difference between a compound Poisson Process and a mapping integrated w.r.t a Poisson random measure? [closed]

What is the difference between $ (1) \sum_{i=1}^{N_{t}}Y_{i} $ and $(2)\int_{[\mathbb{R^{+}}]\times[E] }\eta(e)\tilde{N}(dt,de)$ where $N_{t}$ is a Poissonprocess independent to i.i.d random variables ...
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Help with Poisson Process Question?

I am trying to do the following problem:"Rock tickets are sold at a ticket counter. Females and males arrive at times of independent Poisson processes with rates 30 and 20 customers per hour. If ...
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Laplace fucntional of Bulk Arrival Process

Bulk arrivals. Customers arrive in buses. Buses arrive according to a Poisson process of rate $\alpha$, represented by $\sum_n\epsilon_{\Gamma_n}$. Number of customers in $k$th bus is a random ...
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Poisson Process, infinite servers: Waiting time distribution and expected waiting

Buses arrive at a restaurant as a Poisson process with rate $\alpha$. Each bus contains a random number of hungry customers constituting iid non-negative integer valued random variables. The ...
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2 independent poisson processes: expected time of arrival of one process after the first arrival of the other has occurred

Drug incidents occur in a bathroom as a Poisson process at the rate of $2$/hour and Harry goes in to detect them as a Poisson process at the rate of $1$/hour. Assume that the 2 processes are ...
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Renyi Traffic Model: what process do the final positions of cars form?

At time $0$, cars are positioned along an infinite highway according to a homogeneous PP with rate $\alpha$. Assume the initial position of the $n$th car is $X_n$. Each car chooses a velocity ...
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Customers arrive at a neighborhood grocery store in a Poisson process

Customers arrive at a neighborhood grocery store in a Poisson process with a rate of 5 customers per hour, starting at 8:00 a.m. Upon arrival, a customer remains for Exponentially distributed time ...
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SDE driven by Poisson Process

Suppose that $(N_t)_{t\in \mathbb{R}^+}$ is a Poisson process with intensity $\lambda$>0 and that $a\in\mathbb{R}$ and $X$ being a stochastic process which solves the following SDE:$$dX_t=aX_t^-...
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Poisson process and expected value

Let $N(1);t\leq0$ be a Poisson process with intensity $\lambda >1$. I have $E((N(2)−N(1))(N(4)−N(2))) + E((N(2)−N(1))^2) + E(N(1)(N(4)−N(1)))$ which somehow ends up as $2\lambda^2+\lambda+\lambda^2+...
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Intensity estimation of the marked process.

I observe a sample of the form $𝑆=[𝑆_1,𝑆_2,…,𝑆_𝑁]=[(𝑡_1,𝑥_1),(𝑡_2,𝑥_2),…,(𝑡_𝑁,𝑥_𝑁)]$ where each $𝑡_𝑖$ is an arrival time and 𝑥𝑖 is the amount of money spent by the 𝑖th customer. I ...
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Wrong expected value of sum of Poisson process wait times [duplicate]

Assuming a Poisson process $N_t$ and denoting wait times $S_k$ (i.e. times until the $k$-th jump), I want to find the expected value of their sum: $$ \mathrm{E}\left[\sum_{k=1}^{N_t} S_k\right]\;. $$ ...
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Compound Poisson Process infinitely many jumps?

When I look at the following compound Poisson Process, where $(N_{t})_{t\geq0}$ is a Poisson Process with Parameter $\lambda$ and $\xi_{i}$ are $\mathbb{R}$ valued i.i.d random Variables independent ...
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Alternative definition of the Poisson point process

A Poisson process with intensity $\lambda > 0$ is a random function $t \mapsto N(t)$ with domain $[0,\infty)$ and taking values in $\mathbb{Z}_{\geq0}$ such that i) N(0) = 0; ii) if $s\leq t$ then $...
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Expected Damage per Decade (Poisson Process & Exponential Distrubution)

I should solve the following task: In a certain city, earthquake occurrences are modeled by a Poisson-process with intensity $\lambda $ = 2 per year. The damage caused by such an earthquake, is ...
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Maximum and minimum interval time in Poisson process with fixed number of events

Imagine your favourite Poisson process over a fixed period of time, eg. a particularly high-scoring sports game. Let us say that the parameter $\lambda$ of this process is unknown, but the total ...
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Describing thinning of Poisson processes in notation

How does one describe thinning in notation. For example a question says: Claims arriving under A follow a Poisson process with a rate of 5 claims per day: $\lambda = 5$. Claims arrive independently ...
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Poisson Process Arrival Rate Estimation

Poisson Process Arrival Rate Estimation I think we are supposed to use the fact that Poisson is memoryless but I don't understand how that helps us in calculating the arrival rate.
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Infinitesimal definition of the Poisson process problem

The infinitesimal definition of the Poisson process is given as follows $$P(N(t+h)=n|N(t)=m)= \begin{cases} \lambda h +o(h) & \text{if } n=m+1\\ 1- \lambda h +o(h) &\text{if } n=m ...
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Poisson process working out expected time

The potholes in a road occur as a Poisson process of rate 4 per mile. So, $R(t) =$ "the number of potholes in the first $t$ miles of the road". The situatuion we are given is that "A ...
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1 vote
1 answer
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Relating the definition of a Poisson process with the following definition of a point process

I have difficulties in understanding point process. Here is my problem : We start with the following definition of a point process Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and $(...
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Sum $1/x$ for each $x$ in a Poisson process in $\mathbb{R}$ of intensity $dx$.

Let $N(\omega,A)$ denotes a Poisson random measure on $\mathbb{R}$, for $\omega$ in a probability space and $A\subset \mathbb{R}$, with intensity being the Lebesgue measure. Then, we define a random ...
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Transition probability of a pure death process

Problem: The birth and death process with parameters $\lambda_n=0$ and $\mu_n=\mu, n>0$ is called a pure death process. Find $P_{i j}(t)$. Solution: Since the death rate is constant, it follows ...
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Contribution of independent Poisson Processes within a merged process

Let's say we have a stochastic sequence of events modelled as a Poisson process with a rate of $\lambda$ events per unit interval. A agent tries to take a step of length $\Delta$ units. If an event is ...
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Proof of Campbell's formula for compound Poisson process

I am having difficulty proving the final bit of Campbell's formula, that is, given $C_t = H_1 + \cdots + H_{N_t}$ a compound Poisson process with iid jumps $H_k ~ \mu$ and an independent Poisson ...
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Mean, Variance, and Conditional PMF from Poisson Process

In good years, quarrels between Tolstoy and his wife occurred according to a Poisson process with rate λ = 5 per month. In bad years, it was a Poisson process with rate μ = 10 per month. Suppose each ...
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Marginal of a point process

I am stuck in a problem that involves point processes with exponentially distributed inter-arrivals (or activations), and I identified one of my difficulties in understanding the marginals of such ...
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Which intensity functions are allowed in a non homogeneous Poisson process?

Supose $(X_t)$ is a non homogeneous Poisson process with intensity function $r$. Let $T_1$ be the first arrival time of the process and $R(t) = \int_0^t r$. Then \begin{align*} {\bf{P}}(T_1>t) &...
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Question on Poisson process (my solution so far is included) [closed]

Suppose that, on the day an assignment is due, students hand in the assignment at an average rate of 4 per hour. Suppose the assignment is due at midnight, so there are 24 hours in which students can ...
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2 votes
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How to compute $\sum\limits_{k=0}^\infty \dfrac{\lambda^{2k}}{(k!)^2}$ [duplicate]

This is my exercise Goals occur in a soccer game according to a Poisson Process. The average total number of goals scored in a 90-minute match is 2.68. Assume that two teams are evenly matched. ...
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What is the expectation of a summation where the number of terms itself is a random variable?

I have to calculate the following $$ \mathbb{E} \sum_{i=1}^{N(t)} (t-T_i) $$ where $N(t)$ is a random variable that follows a Poisson process and $T_i$ is the arrival times. Can I do the following: $$ ...
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recovering the poisson process given a generator matrix

We learned about generator (rate) matrices in class, and we showed how to retrieve the generator matrix given some transition matrix that represents a Poisson process. Is there a way to go in the ...
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Power spectrum of Poisson process

Im looking to compute the power spectrum of the time-dependent position of an enzyme that has taken n steps of size "d" at various time points ti (i = 1,..., n): $x(t) = d·\sum_{i=1}^{n} \...
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Linear process of birth using generating function

How do I solve Kolmogorov differential equations for linear process of birth using generating function? By linear process of birth I mean Poisson process for which $$q_{i,i+1}=i\lambda\\ q_{i,i}=-i\...
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Basic Poisson expectation question

I'm trying to grasp poisson processes once again after taking a statistics course a long time ago. I want to find $E(X|X>2)$ where $X \sim poi(3)$ My solution so far: $E(X)=\lambda = rt$ where $r=$ ...
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Poisson Process: number of costumers in a store

Question: Your friend owns a hardware store. On Saturdays, the store is open from noon to 7:00 p.m. During these hours, customers arrive according to a Poisson process at a rate of λ customers per ...
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Poisson Process with conditional probability: independence of λ

Let's say that for $E([0,w), k)$ denotes the event of observing generated observation in the Poisson process k times in the interval $[0,w)$. I know that I can write $$P(E([0,w],k)) = \frac{(\lambda w)...
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Why use infinity as upper bound for Campbell's theorem in this analysis (stochastic geometry)

I am reading a few papers on stochastic geometry analysis of wireless networks and when modeling interference effects at a reference point, the upper bound for Campbell's theorem is set to infinity. ...
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Markov chain extracted from Poisson process

Let $(N_t)$ be a Poisson process. I'm trying to show that, extracting the integer times, I get a Markov process, namely setting $X_n=N_n$ ($n\in \mathbb{N}$), the family $(X_n)$ is a Markov process. ...
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role of the independence assumption for Poisson point processes

A random set $\Pi$ is a Poisson point process with intensity measure $\Lambda$ (a locally finite measure), if For every bounded, measurable set $A$, the number of points in $\Pi \cap A$ is Poisson ...
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An expression for $P(N_s=k|T_n=t)$ where {$N_t ; t\geq0$} is a Poisson process with parameter $\lambda$

Problem Let {$N_t ; t\geq0$} be a Poisson process with parameter $\lambda$ Solve for $P(N_s=k|T_n=t)$ where $T_n$ is the arrival time of the $n$-th event and $s$ and $k$ are positive integers My ...
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Blackwell's argument: Quadratic variation as an upper bound to an expected value in Kingman's book on stochastic processes

Let $S$ be a Borel subset of a complete measurable metric space, and $S^{*}:=S\times(0,\infty)$. There exists a countable family of subsets $B_{1},B_{2},...\subseteq S$ with the property that for any ...
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Example of a "Poisson process" with no piecewise constant trajectories almost surely

Let $T_n$ be an independent sequence of exponential random variables with rate parameter $\lambda$ and $W_n = T_1 + \cdots +T_n$. Define the random variable $N_t(\omega) = \sum_{j = 1}^{\infty} 1_{\{...
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Best Poisson process parameter that satisfies a certain condition.

There is a Poisson process I'm observing for a window $4.5$ units in length. I'm conditioning on the fact that the first event happened within $1.5$ units from the start of the time window. Meaning ...
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How to solve this question about Poisson distribution and does the given probability of women customers affect the question?

Customers arrive at a store at the rate of 10 per hour. Each is either male or female with probability 1/2. Assume that you know that exactly 10 women entered within some hour (say, 10 to 11am). (a) ...
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Modeling the rate at which people leave an office

For clarity, this is the original problem statement in its entirety: People walk into the post office at a rate of 5 people/hour. We consider the following two ways of modeling the number of people ...

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