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]
1414 questions
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Poisson process and interrarrival times
Certain electrical disturbances occur according to a Poisson process with rate $3$ per hour. These disturbances cause damage to a computer.
Assume that a crash will not happen unless there are two disturbances within $5$ minutes of each other.…
Mr. Bromwich I
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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 $$\lim_{n\to \infty }\sum_{k=0}^{n-1}f(t_i)(N_{t_{i+1}}-N_{t_i}),$$
where $\{t_i\}$ is a partition of $[0,1]$ or…
user659895
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Distribution of stochastic integral w.r.t. to inhomogenous Poisson process
My question is related to Distribution of stochastic integral w.r.t. to centered Poisson process[this].
If I have an inhomogenous poisson process with intensity $\lambda(t)$ what is the distribution of
$$X(t) = \int_0^t u(s) dN_s =…
Michael Mark
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How the number of arrivals in the future effects the number of arrivals in the past in a Poisson process?
Let $\{N(t), t \geq 0\}$ be a Poisson process with rate $\lambda$, for $s < t$ find
(a) $E[N(t)|N(s) = 4]$
(b) $E[N(s)|N(t) = 4]$
I solved (a) as $E[N(t)|N(s) = 4] = 4 + E[N(t - s)] = 4 + \lambda(t- s)$ but I do not know how can I solve (b).
P.S.…
errorist
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Customers arrive at a facility according to a Poisson process
Customers arrive at a facility according to a Poisson process $N(t)$ of rate $\lambda = 5.5$ customers/hour. Each customer is admitted to the facility with probability $p=0.6$. All customers, who are not admitted, leave and do not come back. Let…
waterr
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Show that $P(\max_{i=N(s)+1,..., N(t)} (X_i)\leq x)=e^{-(t-s)e^{-x}}$
Let $N$ be a standard homogenous Poisson process, independent of the iid standard exponential claim sizes $\left(X_i\right)$ Define:
$$M_{s,t}=\displaystyle\max_{i=N(s)+1,...,N(t)} \left(X_i\right), \text{ } 0\leq s
Flems
- 416
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Evaluate $P(\bigcap_{t>0}\bigcup_{i=0}^1 \{\lim_{n\to\infty}(N(t)-N(t-\frac{1}{n}))=i\})$ where $N(t)$ is a Poisson process
$N(t)$ is a poisson process with constant parameter $\lambda$, please evaluate
$$P(\bigcap_{t>0}\bigcup_{i=0}^1 \{\lim_{n\to\infty}(N(t)-N(t-\frac{1}{n}))=i\})$$
It's easy to derive $P(\{\lim_{n\to\infty}(N(t)-N(t-\frac{1}{n}))=0\})=1$, but I'm…
Oolong Milktea
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Inter-arrival time distribution of a Poisson process
I'm actually looking at a rigorous way to prove that if $\{N_t\}_{t\geq 0}$is a Poisson process of rate $\lambda >0$, then the inter-arrival times $(H_n)_{n\in \mathbb{N}}$ are independently and identically distributed exponential random variable…
Arthur Serres
- 103
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Poisson process probability calculation
I have the Poisson process $\{N(t)\}_{t\geq 0}$ with rate $\lambda=2$. Given that four events occur during the time interval $[0,2]$, what is the probability that the first event occurs before time $t=1$?
From what I understand, I need to calculate…
TK99
- 427
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Fishing Poisson Process
Jack likes to go fishing. While waiting for the fish to bite, he formulates the following model for the process: fish bite according to a Poisson process with intensity 4 bites per hour. Biting fish are caught independently, and on average only one…
ttucker34
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Recursive Poisson Process?
I would like to know if there is a name for a type of process that works as follows: Suppose we have a Poisson process with rate paramter $\lambda$. At each new arrival at time $t$ of this process, a new sub-process begins, which is also Poisson…
theQman
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Find Probability of Poisson process
I was doing some questions from my textbook and came across this problem that I'm stuck on how to solve:
Skiers arrive to a top of slope one at a time using a ski lift.
The number of arrivals follows a Poisson process with rate λ = 20 arrivals per…
Raaa
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Conditioning in a Poisson process.
Question. Let $X(t)$ be a Poisson process modelling the arrival of alpha particles at a detector after $t$ hours with a rate of $2$ per hour. Let $Y(t)$ be a Poisson process modelling the arrival of beta particles at a detector after $t$ hours with…
thesmallprint
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time between poisson events
The number of calls during a one hour shift period has a Poisson distribution with a mean of 8. This question is: let Y be the time in hours between the arrival of the first and second call. Find P(Y< 0.1).
The density function of the time between…
Denson
- 91
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How is this a "non-homogenous Poisson process"
The question is:
The number of car accidents an insured person faces depends on their experience. The rate at which Liam will have car accidents is given by:
$$R(e) = \frac{(e+0.5)^{-2}}{200},$$
where $e$ is the experience in years.
I am required…
