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birth-death process

This is from notes I have 2 questions: why $\lambda h$ or $ \mu h$ is the probabiity of one unit change? And let's say the birth rate is 1 per min, and the h is 2 mins, obviously there are two ...
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1answer
25 views

Conditional distribution (on N) of arrival times in a nonhomogenous poisson process

Conditional on $N(t)$, given some $\lambda(t)$ characterizing some Nonhomogenous poisson point process, the distribution of an arrival time $t_i$ is $\lambda(t_i)/\int_{A}\lambda\left(t\right)dt$ ...
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30 views

Probability Distribution for Inter-arrival Time

I'm trying to build a simulation for a quality control process, where quality analysts inspect the product and report faults if they found any. I have a dataset of this bug reports, so I'm trying to ...
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0answers
12 views

how to incorperate two independent variable into function rexp in R?

A simple electronic device consists of two components which have failure times which may be modeled as independent exponential random variables. The first component has a mean time to failure of 3 ...
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5 views

About Garman's inventory model

In Garman's inventory model (http://www.sciencedirect.com/science/article/pii/0304405X76900064), buying order and selling order are Poisson processes with order size = 1. Buying price and selling ...
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6 views

Is it possible to derive a relation between parameters in Poisson process representation of extremes and parameters in GPD model?

I want to derive the theoretical relation between the parameters in a point process model for extremes and the parameters in the GPD model for extremes. I'm following Coles - An introduction to ...
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1answer
125 views

Age and residual life time of the Poisson process

Original Question Let $N(t)$ be a Poisson process with intensity $\lambda$. Let $T_1<T_2<...$ be the occurrence times. Let $T_0=0$. For any $t>0$, define the $age$ random variable to be $...
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193 views

Predicting intensity of Poisson process, given event data

I have a dataset of events: each row is an event, and each column is a feature. There are millions of events and several dozen features. The features are mostly numerical (a few are categorical and I ...
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1answer
13 views

Question about marked poisson process

Let's say I have a Poisson point process on $\left[0,T\right]$ with rate $\lambda\left(t\right)=2t^2$. Suppose I attach a mark $m_t$ to each point $t$ of the process such that $m_t\sim N\left(t,1\...
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0answers
12 views

Modeling Arrivals With a Time Limit

I have some data (sample here): ...
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1answer
52 views

Problem on Poisson Process

I am doing some problems related with the Poisson Process and i have a doubt on one of them. The problem is stated as follows: A doctor works in an emergency room. The emergencies arrive according a ...
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1answer
78 views

Find the distribution of $ N = \min \left\{k: \prod_{i = 1}^{k}U_i \lt .6\right\}. $

I'm cross-posting this from math.SE because it's not getting any love over there. However, if that's considered heresy, I can delete the posting over there. The Statement of the Problem: Let $ \{ ...
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1answer
42 views

Maximum value in Poisson process investigated using scan statistics

We have process where events are occurring at a rate of $B$, where the distribution of events in a fixed time frame can be described using Poisson statistics. Thus, the events can be modeled using a ...
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1answer
18 views

Decomposing multiple poisson process

Assume a time series composed by many recurring events coming from many different poisson process each with a different rate. Lets assume for simplicity no overlap between events. Is there any math/...
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0answers
24 views

Testing for significant difference between two conditions for a poisson process with multiple replicates

I am a biologist rather than a statistician, so I apologise for any misuse of terminology. Happy to clarify if needed, and I am comfortable using R for statistical tests. I recently conducted an ...
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4 views

How to do regression on event density and size?

I have events with time and size. It might look like $(t_1, x_1), (t_2, x_2), \ldots $. What I really care about is average total size per time $\lambda =\frac{\sum_{k=i}^j x_k}{t_j-t_i}$. The $x_i$ ...
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1answer
80 views

Time up to $n$th event in Poisson process distributed as $\frac{1}{2\lambda} \chi^2_{2n}$

Let's assume that a number X of some events over time $t$ is modeled by Poisson distribution with rate $\lambda$ (here, it's rate, not mean): $$ X \sim Poisson(\lambda \cdot t) ~~~~ (\lambda t ~\text{...
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45 views

Poisson process for queuing problem

I posted it incorrectly on a different thread so I'm reposting it here. I'm working on a problem and would love to get any advice - patients come into a clinic according to a Poisson process with ...
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65 views

R Poisson process simulation for a queuing problem [closed]

I'm working on the problem - patients come into a clinic according to a Poisson process with time parameter 10 minutes starting from 9 am until 4pm when the clinic stops admitting new patients. There ...
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245 views

Switch from Modelling a Process using a Poisson Distribution to use a Negative Binomial Distribution?

We have a random process that may-or-may-not occur multiple times in a set period of time $T$. We have a data feed from a pre-existing model of this process, that provides the probability of a number ...
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1answer
97 views

Probability of a rare event

Let's say I consider an event rare if it occurs no more than once in 90 days. Assuming everything is random and independent, If I see this event on day 3 of the observation, what is the probability ...
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61 views

Help with Poisson Process

I'm going to repost this here since my questions never get answered on mathstackexchange. It might be better suited to this location, as well. At the end of the workday, I add an amount between 0 and ...
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1answer
111 views

Exponentially decaying integral of a Poisson process

Suppose that $X_t$ is the set of times of the events of a Poisson process with unit rate after $t$ seconds. (In other words, $X_t$ is a set of $N$ uniformly distributed points over $[0,t]$ where $N$ ...
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1answer
30 views

Poisson distribution of rain storm arrival

I know what poisson distribution means.But I can not just understand how rain cells or rain storm arrivals is poisson process? looking for simple explanation. Thanks in advance
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1answer
15 views

Odds of specific generated population of exponential distributed stochast

I'm trying to generate a sequence of samples using an exponentially distributed stochast, i.e., making a Poisson arrival process. In my specific case I generate 337 samples using a mean inter-arrival ...
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24 views

Confidence interval for non-homogeneous Poisson process where lambda is fourier

I have a non-homogeneous Poison distribution $X = \{ x_1, x_2\, .., x_n\}$ where: $$\lambda(x) = \exp(a_0 + \sum_{z=1}^{Z}(b_z \sin(2 \pi x \frac{z}{Z}) + c_z \cos(2 \pi x \frac{z}{Z})))$$ $$\Lambda=\...
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1answer
60 views

Expected Value in Poisson Point Process with Prior Knowledge

I have a setup with a homogeneous Poisson Point Process (PPP) of intensity $\lambda$ in $W \subseteq \mathbb{R}^d$ and a set $A \subseteq W$. I'm looking for the expected value of points in set $A$, ...
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0answers
26 views

Question about M-step for bimodal Poisson Proces

all I have difficulty in deriving the result for the M-step of EM of the bimodal Poisson as shown in paper, Byers, Simon, and Adrian E. Raftery. "Nearest-neighbor clutter removal for estimating ...
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2answers
80 views

distribution of the last arrival in poisson process

Consider a Poisson process with rate $\lambda$ and let $L$ be the time of the last arrival in the interval $[0,t]$, with $L=0$ if there was no arrival. How can I prove that t-L has exponential ...
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36 views

How do we calculate this transition count probability?

Note: This is from the proof of prop 3.6 here. We want to calculate the probability of one transition between states $i$ and $j$, ending at state $r$, and starting at state $l$, in a time slice of ...
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32 views

Expected number of times two Poisson process events occur on the same day

So I have set up a Poisson Process N(t) with parameter L (events/year). I want to find the expected number of times over a 3 year period that 2 events occur on the same day. My approach: First ...
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1answer
20 views

Wait time for non-homogenous Poisson processes

Is the wait time between events for a non-homogenous Poisson process still exponentially distributed
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26 views

Transformation of growth function with two pulses

I'm trying to linearize the following exponential growth function: $$ y = ae^\frac{t}{b}+(1-a) e^\frac{t}{c} $$ To preempt any question regarding why I'm doing this: I'm trying to regress this ...
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0answers
41 views

Prediction Error for a Poisson Process

So I have a fleet of cars. I've sampled their breakdown rates (per hour driven) for the last 10 years. Their breakdown rate follows the gamma distribution. I am trying to do two things. Predict ...
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0answers
126 views

Autocorrelated Inter-arrival Times of Extreme Events

I'm using a bunch of techniques and methods from Extreme Value Theory to analyze my data. I have a time series representing the number of events happening in a given day. The time series is unequally ...
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0answers
18 views

Point process models

Two of the popular models for analyzing point process data are Cox and Hawkes processes. My question is how do we compare the statistical properties of these two processes as they both can be ...
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0answers
42 views

Estimate lambda for panel count data

I have panel count data for $F$ firms across $I$ years, so observe counts $C_{f,i}$ for $f \in \{1,...,F\}$, $i \in \{1,...,I\}$. I want to model the data as a poisson process. With increasing $C_{f,...
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0answers
31 views

Pmf of compound Poisson process

Can I obtain an analytic expression for pmf of compound Poisson process? $Y_t = \sum \limits_{i=1}^{Xt} D_i$ where $X_t$ ~ Poisson($\lambda$) and $D$ ~ Geometric($\rho$)
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46 views

Poisson vs. Gaussian in Geomagnetic Data

I've been studying geomagnetic signals using a threshold approach to detect pulse events in the data. The question here is what is the significance of the crossover of stddev and mean as the ...
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1answer
53 views

Alternatives to a one-dimensional Poisson process [closed]

Say I have "arrival" times in what may or may not be a Poisson process. I can think of at least three ways in which it can deviate from a Poisson process: Clumping. One arrival is likely to be near ...
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0answers
26 views

Prove that $N(\tau),V_t,Z_2,\ldots$ are independent in Poisson process

We define a Poisson process is a renewal process in which the interarrival intervals $X_n$'s have an exponential distribution with parameter $\lambda$. Denote $N(t)$ is the number of arrivals in $(0,t]...
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0answers
149 views

estimate confidence interval for poisson process

I would like to know how I can estimate the confidence intervals for poisson process distributed variables. I have a pandas dataframe with a column of trials and a column of successes. I want to ...
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0answers
74 views

Fit to a non-homogeneous Poisson process

I have a sequence of event timings, $t_1, t_2, t_3, ..., t_n$ where there are a total number of $n$ events happening. For example, they are the timings when 911 was called in a city. I know these ...
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1answer
1k views

Poisson Process in R from exponential distribution

I am trying to simulate a poisson process sample path in R by starting off with exponentially distributed random variables. For example, for a value of $\lambda=0.5$, I can generate 500 samples and ...
2
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1answer
108 views

Using Poisson process model for prediction?

Suppose some event $X$ occurs on average $10$ times per minute. The events are independent of each other. Now, if I have understood correctly, this can be modeled as a Poisson process, and I can ask ...
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0answers
19 views

Overlapping interval of a Poisson arrival process

Calls arrives according to a Poisson arrival process with rate lambda = 15. Find E(N(2,4]N(3,5]) My thoughts: E(N(2,4]) = E(N(3,5]) = lambda * t = 15 * 2 = 30 However, I cannot figure out the next ...
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1answer
19 views

Is there a family of processes centred on the Poisson process?

I am looking for a model, characterized continuously by a single parameter, to describe the arrival times of buses with unit expected interarrival time. At one extreme of the parameter (say $\theta=1$)...
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46 views

A math proof within a question about homogeneous Poisson process

We know that a homogeneous Poisson process is a process with a constant intensity $\lambda$. That is, for any time interval $[t, t+\Delta t]$, $P\left \{ k \;\text{events in}\; [t, t+\Delta t] \right \...
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57 views

Help me out with real-life Gamma distributions

I'm working with some data relating to maintenance and parts failures. I've got a pure math background with only a little bit of probability, and I'm currently learning by doing. I've got a list of ...
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79 views

Understanding Poisson Point Process [closed]

Could someone explain the Poisson Point Process? Is it simply an Integration over a Poisson Process for higher dimensions? If so, how do you derive the function, say in two-dimensional space, and 3-...