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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Waiting time in Poisson process
Let $\{X(t) : t \geq 0\}$ be a Poisson process with rate $\lambda$, and let $W_n$ denote the waiting time for the $n$-th event. For $s \geq 0$, determine $P( W_{X(t)} \leq t+s)$ and $P( W_{X(t)+2} \...
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14 views
nearest neighbor for a Poisson process in the plane
I do not know why the term called "nearest neighbour" while in all the place
treat it as all the neighbours within a certain area like here
Doesn't $P(X<r)=1-P(X>r)=1-P(U_r=0)$ mean the ...
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9 views
superposition of infinitely many poisson processes
I know that the superposition of two Poisson process with rates $\lambda_1$ and $\lambda_2$ is again a Poisson process with rate $\lambda_1+\lambda_2$. Thus this process has interarrival times ...
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8 views
Poisson Process/Inter-Arrival Time Question
This question is, "People arrive at a train station in accordance with a Poisson process with rate lambda. At time 0, the train station is empty. At time 10, the bus departs. Calculate the expected ...
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8 views
Show that the number of elements in a given set is a Poisson process.
Let $\quad \{ \nu_n, \xi_k^n \}\in \mathbb N$ be a family of random variables. $\quad \nu_n \sim Poiss(\lambda), \quad \xi_k^n \sim Unif(n-1,n].$
Let $\quad N_t=|\{ (k,n): k \le \nu_n, \quad \xi_k^...
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14 views
Probability of Ruin at the first claim
The number of claims $n \sim Po(\lambda)$, and let $X_n$ denotes the claim amounts of a claim which are all iid and they follow a $Exp(1)$ distribution.
Assume the initial surplus is $U$, and the ...
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67 views
Double integral over square converging to $0$
I am struggling to solve the following problem as part of a bigger project that I am working on.
Let $\mathcal{S} \subset \mathbb{R}^2$ be a square of length $\sqrt{n}$ centered at the origin, $f:\...
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21 views
Byzantine Fault Tolerance Threshold of Bitcoin: 1/2 or 1/3?
Although, this question is related to the Bitcoin network; however, its calculation is more relative to the Mathematics. So, let's bring it up here:
According to this answer: https://bitcoin....
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1answer
34 views
Poisson process that degrades uniformly
Given that people arrive according to a poisson process with rate 4 per minute and stay for $X$~Unif[0,10] number of minutes independently of the arrival times. What is the mean and variance of the ...
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8 views
Rate parameter in Poisson process - confusion regarding terminology
I am trying to understand a derivation of maximum likelihood estimation of intrinsic dimension from a Poisson point process given here https://wiki.math.uwaterloo.ca/statwiki/index.php?title=...
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25 views
The expected distance from a point to its neighbors in homogeneous spatial poisson process
Let $n$ point are distributed as per a homogeneous spatial Poisson process of rate $λ$ in a square of side $2a$, and $R$ be the distance from a point to its neighbors within a distance $r$. What is ...
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33 views
How to calculate realization of a random process?
I am newly learner of subject of stochastic processes and my mind is full of questions. Hopefully I can ask one of them in a correct way.
Suppose that X is an random variable which follows ...
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39 views
If $N$ is a Poisson process, then $\operatorname P\left[\exists t>0:N_t=n\right]=1$
Let $(N_t)_{t\ge0}$ be a Poisson process with parameter $\lambda\in[0,\infty)$ and $n\in\mathbb N$. How can we show that $\operatorname P\left[\exists t>0:N_t=n\right]=1$?
Assume for the moment ...
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1answer
33 views
The average time before a person find their group
Imagine there are $N$ people throwing a party. For any two of them, the time before they meet each other and stick together thereafter is independent, and obeys an exponential distribution whose $\...
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21 views
Order non-homogeneous Poisson process rate functions by specific (end) time
The broader problem I am trying to solve is the following:
Given $k$ lists of arrival times (all times less than or equal to some end time $T$) coming from $k$ non-homogeneous Poisson processes with ...
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2answers
63 views
Poisson Process: Time until next arrival
Question:
Suppose that busses arrive at a bus stop as a Poisson process with rate $\lambda$ starting from time $t=0$ (that is, the interarrival time between busses is exponentially distributed with ...
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16 views
Compensator of a non-homogenous Poisson process
Given a non-homogenous Poisson process $T_n$ with intensity $\lambda(t)$ and the compensator $\int_0^t\lambda(t)ds$
How can I show that $\int_0^t\lambda(s)ds$ is predictable?
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8 views
predictable projection of non-homogeneous poisson process
How to prove that the compensator of a non-homogeneous poisson process is the predictable projection of this non-homogeneous poisson process?
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15 views
Poisson Process with Stationary Arrival Rate - Conditional Arrivals
I am having trouble with the following,
We have a Poisson process that operates 24 hours per day with an arrival rate $\lambda = 3$ per hour. Suppose an observer arrives at 3pm. By 5pm, the observer ...
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0answers
18 views
Construction of probability measure for Hawkes process
How to construct a probability measure for the Hawkes process?
Like here
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0answers
29 views
Distribution of inter-arrival time of non-homogenous Poisson Process
What is the distribution of inter-arrival time of a non-homogenous Poisson Process?
In other words, if $T_n$ is a non-homogeneous Poisson process with intensity $\lambda(s)$, and $S_{n+1} = T_{n+1} - ...
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27 views
Conditional law of Poisson process
1)What does the notation for $G_n$ and $H_n$ mean?
2)How would $G_n$ and $H_n$ look like for a Poisson process?
3)How to show that $v$ would be the compensator of a Poisson process?
Thanks!
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18 views
Counting process with independent, stationary increments is Poisson
Suppose that $L_t$ is a counting process, i.e.
$$
L_t= \sum_{i \in \mathbb N} 1_{T_i \le t}
$$
for a collection $(T_i) _{i \in \mathbb N}$ of stopping times.
I have often seen the claim that if $L_t$...
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2answers
76 views
Interarrival Time Distribution of a Poisson Process
For a Poisson Process with parameter $\lambda$ restricted to the interval $[0, 1]$, what is the probability that at least one of the interarrival times (including the time between $0$ and the first ...
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15 views
proof of the expected value for two Poisson point process
Consider $H_1$ and $H_2$ as two independent homogeneous Poisson point process in $\mathbb{R}^2$ space of intensities $\lambda_0$ and $\lambda_1$, respectively. We divid a square area into some region ...
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12 views
Counting Process Conditioned on Random Rate
Having some trouble figuring out this problem. Say you have a counting process $N_t$ where the rate is a random variable $\Lambda$ which is exponentially distributed with rate $\alpha$. I'm trying to ...
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1answer
22 views
Poisson Process Conditional Expectation
Given $X_t$ a Poisson process such that $\lambda = 1$ find $E[X_1\mid X_2]$ and $E[X_2\mid X_1]$.
The first one is pretty straight forward since we have $E[X_2 - X_1] = E[X_1] = 1$ so then we get $E[...
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17 views
Escaping probability for a poisson random walk
Poisson random walk: Let independent random variables $Z_i \sim Pois(\lambda)$. Consider random walk
$
S_n = \sum_{i=1}^{n}X_i,
$
where
$$
X_i = \begin{cases}
Z_i &\text{w.p}\; p\\
-Z_i &\...
3
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1answer
104 views
Poisson point process in 2D with reflecting boundaries
Consider a point process $\{(X_n,T_n)\}$ on a plane $[0,1/\lambda]\times\mathbf R^+$, generated from a Poisson point process $\{T_n\}$ with rate $\lambda$ on $\mathbf R^+$ (i.e. $(T_n-T_{n-1})$ is iid ...
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1answer
18 views
Why is this inequality true in proof of strong law of large numbers for renewal processes
Let $\{N_t\}_{t\ge 0}$ a renewal process with intensity $\lambda>0$. Then $\lim_{t\to\infty} \frac{N_t}{t}=\lambda$ a.s.
Here $W_i$ denotes the waiting time of $N_t$ and $T_n:=\sum_{i=1}^n W_i$ ...
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1answer
24 views
Conditional Expectation of Poisson Process Interarrival Events
I'm having some trouble with something my professor said would be on our exam. For a Poisson process $N_t$ with interarrival times $X_i$, how is it that you find $E(X_i | N_t = n)$ (assuming $n\ge i$)?...
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1answer
70 views
Poisson Point Process On Real Number Line
Let $\mathbf{\Pi}$ be a homogeneous Poisson point process on the real number line with intensity $\lambda$. Let $r^{+}$ denote the distance from the origin to the closest point of $\mathbf{\Pi}$ on ...
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1answer
56 views
The sum of the total number of service representatives who spoke with all the customers who called during a minute.
I trying to solve this problem, but I can not tell if what I am doing is right,
At some company, the customer support department gets called by $5$ customers per minute independently.
The ...
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1answer
43 views
If the inter-arrival times of customers are i.i.d. exponential distribution, is it necessary that the number of customers is a Poisson process?
Suppose customers arrive with time interval $U_i$ i.i.d. $Exp(\lambda)$, therefore,
$$F(U_i\le t)=1-e^{-\lambda t}$$
The arrival time of customer $i$ is
$$T_i=\sum^i_{j=1}{U_j}$$
The number of ...
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1answer
32 views
Find expected value of compound Poisson process
We have compound Poisson process with $\lambda = 3$ and jumps $D_{j}$ of size $1$ or $2$, where $P \{ D_{i} = 1 \} = 0.25$. Find mean value of this process in $t = 10$.
Mean value of compound Poisson ...
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19 views
expected value from some points in continuous homogeneous spatial Poisson point process
Let $n$ point are distributed as per a homogeneous spatial Poisson process of rate $λ$ in a square of side $2a$, and assume that $4$ fixed points are located at $(a/2,a/2)$, $(-a/2,a/2)$, $(a/2,-a/2)$ ...
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14 views
Poisson point process representation
Let $\Pi: ( \Omega, \mathcal{F}, \mathbb{P} ) \rightarrow \mathbb{R}^d$ be a Poisson point process.
We know that $\Pi_0=\{\left \| X \right \|, X\in \Pi\}$ is a Poisson point process on $\mathbb{R}_+$...
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7 views
First event distribution of a non-homogenous Poisson Process when the event is not bound to occur
Let $N(t)$ a non-homogeneous Poisson Process with rate $r(t)$. If the rate sum up to infinity, i.e. $\int_0^\infty r(u)du=\infty$, the first event distribution, $f_{T_1}$ can be expressed as:
\begin{...
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2answers
51 views
Probability that $25$ calls are received in the first $5$ minutes.
Calls are received at a company according to a Poisson process at the
rate of 5 calls per minute. Find the probability that $25$ calls are
received in the first $5$ minutes and six of those calls ...
2
votes
2answers
78 views
Show that the expected total present value of the bonds > purchased by time $t$ is $1000\lambda(1-e^{-rt})/r.$
Investors purchase $1000$ dollar bonds at the random times of a
Poisson process with parameter $\lambda$. If the interest rate is
$r$, then the present value of an investment purchased at time $t$ ...
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1answer
33 views
Probability that at least one other vehicle arrives between third and fourth var arrival.
Starting at $6$ a.m, cars, buses and motorcycles arrive at a highway
toll booth according to independent Poisson processes. Cars arrive
about once every $5$ minutes, buses about once every $10$ ...
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1answer
31 views
Let $(N_t)_{t\geq 0}$ be a Poisson process with parameter $\lambda=2$ find $\mathbb{E}[N_3N_4].$
Let $(N_t)_{t\geq 0}$ be a Poisson process with parameter $\lambda=2.$
Find $\mathbb{E}[N_3N_4].$
The solution here is
\begin{align}
\mathbb{E}[N_3N_4]&=\mathbb{E}[N_3(N_4-N_3+N_3)]\tag1\\
&...
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0answers
62 views
Finding expected value of a stopping time dependent on a Poisson process and a variable $n$
Situation:
We have that $\{W_t,t \geq 0\}$ is a Brownian motion and $\{N_t,t\geq 0\}$ is a Poisson process such that $N_t$ follows a Poisson
distribution with parameter $t$. This process is ...
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1answer
57 views
How to solve this problem with Poisson distribution
Problem:
A store owner observes that there are $3$ (in average) customers visiting the store per hour. He wants to find the probability that there are at least $1$ customer visiting his store in $...
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1answer
72 views
Expected value of Brownian motion at a time decided by a rate one Poisson process.
Situation:
We have that $\{W_{t},t\geq 0\}$ is a Brownian motion and
$\{N_{t},t\geq 0\}$ is a Poisson process such that $N_{t}$ follows a
Poisson distribution with parameter $t$. This process ...
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2answers
34 views
What is the limiting probability that there are n people in the facility?
Customers arrive at a store at a Poisson rate of λ and there is a single server with rate μ. The arrival and service times are independent random variables. Customers leave the facility immediately ...
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1answer
22 views
independent Poisson processes probability
I have been given this question to solve
The numbers of claims to an insurance company from smokers and nonsmokers follow independent Poisson processes. On average 4 claims from nonsmokers and 6 ...
1
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1answer
16 views
Poisson process: finding probability of 1 count in an interval given that 0 counts happen in a subinterval
This was in my exam today and I'm not sure what's the correct answer.
Let's say that the number of people that enter into a store in the interval $(0,t]$ (in hours) is a Poisson process where $30$ ...
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1answer
32 views
A question about Poisson Process: operating events from different sample spaces?
The following proof, that how to derive Poisson Distribution from a Poisson Process, is from my textbook, Elementry Probability Theory(Fourth Edition), written by Kai Lai Chung, Farid AitSahlia.
...
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39 views
Question related to Poisson process
Suppose busses arrive at a bus stop either with an inter-arrival time of exactly 1 min or with an inter-arrival time of exactly 10 mins. Suppose the 1 min inter-arrival times occur with probability 2/...
