For questions about the theory or applications of the Poisson process, one of the most widely applied point processes in statistics and elsewhere.
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Simulate an arrival time - Cox Process?
If I simulate arrival times using a Poisson process where the input to the Poisson process is also a Poisson process.
Input arrival rate = L
, and then U1 and U2 are random draws from a uniform ...
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0answers
20 views
Point process - intensity function vs probability density function
Suppose we have a point process in $\mathbb{R}$ with intensity $\lambda(x)$. Then, for a given compact set ${ S}$ we have
$$\Lambda({ S})=\int_{\rm S} \lambda(x) \, dx,$$
where $\Lambda({ S})$ is ...
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43 views
Probability of an event with multiple conditions
Could you inform me please, how can I calculate conditioned probability of several events?
I have 3 events A, B, C; I know P(B|C) and I want calculate P(A|B,C). Is it possible?
In the special case ...
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6 views
fix the number of class in random poisson simulation
I want to simulate a poisson distribution in R with rpois and in my generated vector I want to have a fixed number of level
example
...
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15 views
LRS for poisson process
How to use likelihood ratio statistics test for Poisson process with different intensity function model?
Model 1: $ \lambda(t)=\lambda$
Model 2: $ \lambda(t)=\exp( \beta_1 + \beta_2 t)$
I think the ...
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26 views
Does exponential waiting time for an event imply that the event is Poisson-process?
Say I have a process, $\{N_t : t \ge 0\}$, which denotes the number of the event that occurred until the time $t$.
And let me define $W = \min \{t : N_t = 1\}$ which is denotes the time until the ...
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1answer
597 views
Nonhomogeneous poisson process simulation
I've been looking at ways to generate a Nonhomogeneous Poisson Process (NHPP) including the nonlinear time transformation (using a rate-1 process and inverting the cumulative rate function). I've also ...
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1answer
34 views
Conditional expectation of Poisson process given number of events
Let $\{N(t), t\geq 0\}$ be a Poisson process with rate $\lambda$, $S_n$ the instant of the $n$-th arrival and $T_n$ the $n$-th interarrival time, that is, $T_n = S_n - S_{n-1}$, $n \geq 1$.
Now ...
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1answer
42 views
Poisson process and queuing system
In a queuing system, the inter-arrival times are known to be exponentially distributed.
My textbook states: "It can be shown that, if the underlying distribution of inter-arrival times { T1, T2, ..., ...
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1answer
237 views
Non homogenous Poisson process with simple rates
I am trying to stimulate number of claims in the next 12 months using a non-homogeneous poisson process. The rates are:
...
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2answers
416 views
Trajectory of homogeneous poisson process
I am trying to simulate number of claims in next 12 months using a homogeneous poisson process following the R codes:
...
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1answer
34 views
Problems on Poisson process
Consider a homogeneous Poisson process with inter-arrival times $T_i$, which follows the exponential distribution with rate $\lambda$. Let $N(t)$ denote the number of arrivals by time $t$.
Suppose I ...
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14 views
Error prediction for poisson process
A Poisson process has rate of N per unit length per unit time. The events occur uniformly on the x-axis but I always have measurement error of greater than 1/N. I was wondering is it possible that I ...
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1answer
91 views
Gamma Conjugate Prior & Poisson Process
I am analyzing daily data transaction data.
I am assuming that
The number of transactions in every day of length t has the Poisson distribution with mean λt
The number of transactions in evert ...
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1answer
24 views
Poisson process (find mean) - solution check
Mary receives 22 messages in 5 hours according to a Poisson Process. What is the mean number of messages Mary receives in an hour?
Seems like a simple question, so just to confirm: Mean = 22/5
Isn'...
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40 views
Binomial/Poisson Problem - Pizza Orders
Suppose that the number of orders per hour at a pizza shop follows a Poisson process with rate 5 per 30 minutes. Suppose that the pizza orders are large with probability 2/3, small with probability 1/...
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1answer
25 views
Poisson/binomal problem - car speeding [duplicate]
A certain police officer stops cars for speeding. The number of red sports cars she stops in one hour is a Poisson process with rate 4, while the number of other cars she stops is a Poisson process ...
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1answer
49 views
Conditional probability with poisson (can this be solved by binomial?)
Pizza orders arrive according to a Poisson process of rate $20$ per hour. Orders are independently for a vegetarian pizza with probability $\frac14$ , and for a meat pizza with probability $\frac34$.
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25 views
Probability of stale cache hits w/ exponentially distributed read and write inter-arrival times
I am trying a create a model for a very simple system which consists of a server and 2 clients: R and W. The server stores a piece of data D and hands out leases for caching D on the clients. The ...
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1answer
19 views
Data of daily counts modelled as Poisson process: should it be compound?
I understand that one of the basic assumptions of a Poisson process is that in a small enough interval, the probability of more than one arrival is negligible.
In my case, I have data that shows the ...
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38 views
What to do with zero-counts when fitting Poisson distributions? [closed]
Suppose we have events that occur in time following some compound Poisson process. For example, let's say the rate parameter is such that an event occurs every 30 days, on average. If we have 10 years ...
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1answer
90 views
MLE for a homogeneous Poisson process?
If we have a data set consisting of event times $\{t_1, t_2, \ldots, t_N\}$ and would like to model this as a Poisson process with intensity $\lambda$, how do we do it? Intuitively, I would expect ...
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10 views
Parameter estimation for NHPP for arrival series
I have multiple data sets (about 15) that describe a process of arriving customers to a shop. When plotted it's pretty clear that over time the rate of customers arriving decreases. However the time ...
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24 views
Poisson processes or regression for lambda
I am asked to solve a problem where I have a machine producing toys in tho slots and want to predict number of faulty toys. The data is like this:
...
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1answer
28 views
What is mean by the term “constant rate” in Poisson distribution?
I have difficulty in understanding the assumptions of Poisson distribution,
one assumption is the rate at which the events occur in the time interval is constant. What is the meaning of that phrase? ...
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11 views
Compute the covariance of random variables in a discrete-time queueing process
Background
Suppose we have $Z$ particles at time $t=0$, with $Z\sim\text{Poisson}(\lambda)$. At each time $t=0,1,\dots,$ a particle either expires or continues to exist. Suppose that for each $t=0,1,\...
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38 views
Find unbiased estimators for $\lambda$ and $\lambda^2$.
For the spatial homogeneous Poisson process, find unbiased estimators for $\lambda$ and $\lambda^2$.
Attempt: Since the homogeneous Poisson process is over an area, how i would i go about ...
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141 views
Prove that the interarrival times of a Poisson Process are all indipendent and identically distributed
{$N_t$} with $t\in \mathbb{R}$ is a Poisson process with intensity $\lambda \in \mathbb{R^+}$, so that
1) $N(0)=0$,
2) {N(t) is with indipendent increments and omogeneous increments and
3) $\...
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2answers
112 views
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0answers
51 views
How do I compare two event rates?
Say I have two event streams, modeled as Poisson processes. These streams omit events at rates $\lambda_1$ and $\lambda_2$ respectively, so the count of observed events in $t$ time is $\lambda_1t$ and ...
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67 views
Waiting time paradox - rigorous math description
I was reading about the waiting time paradox. I can understand that from the memoryless property of the exponential distribution, if the events occur with rate $\lambda$ then the average waiting time ...
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0answers
13 views
How to choose time bin width for Poisson Process?
Newbie here - just started learning more about poisson distribution. I have a time series of occurrence of events, and I thought this might fit a Poisson process. Before I do hypothesis testing, I ...
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0answers
18 views
Compound Poisson process for demand
I have a demand pattern for a service part. Demand event rate of this part is Poisson distributed. Demand of the part is 3 times in a year. So the demand event rate is 0.25/Month. Each demand ...
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1answer
63 views
Normal approximation on (what it looks like) a poisson
I am self-studying inferential statistics from Larson's introductory textbook named: "Introduction to Probability Theory and Statistical Inference" (1st edition John Wiley & Sons.).
I came ...
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1answer
56 views
Poisson Process - Determining Rainfall Accumulation
Disclaimer - I'm not a statistician. Most of my knowledge on the subject is self-taught.
I'm trying to grasp how continuous random variables can be used in conjunction with a discrete Poisson process....
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1answer
26 views
Poisson Process: Probability of 2 arrivals by time 1 given that there are 6 arrivals by time 4?
Let $(N_t)_{t \ge 0}$ be a Poisson process with parameter $\lambda = 1.5$. Find the following:
(c) $P(N_1 = 2 | N_4 = 6)$
I know how to do it if the times are reversed (it simply uses the ...
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0answers
36 views
Poisson Process: In a small parliamentary election, votes are counted according to a Poisson process at the rate of 60 votes per minute
In a small parliamentary election, votes are counted according to a Poisson process at the rate of 60 votes per minute. There are six political parties, whose popularity among the electorate is shown ...
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1answer
62 views
Poisson Process: Computers in a lab fail, on average, twice a day, according to a Poisson process
Computers in a lab fail, on average, twice a day, according to a Poisson process. Last week, 10 computers failed. Find the expected time of the last failure, and give an approximate time of day when ...
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1answer
42 views
Definition of (contiuous-time) Markov chain transition rate
Suppose that the rate at which a Markov chain leaves state i at some time t is $\lambda_{i}$.
I.e., What is the rate at which $X_{t}$ leaves state i.
Then, $\lambda_{i} = \sum_{j\neq i}q\left ( i,j \...
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1answer
182 views
PGF of a Poisson Process
I'm trying to find the probability generating function of a general Poisson Process and am a little stuck.
The PGF is defined as $E(s^{N_t})$, and I know that the density function of $S_n$ is:$$...
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1answer
30 views
Limiting fraction of number of arrival while another event is occurring
A light bulb burns for an amount of time
having distribution F with mean μF then burns out. A janitor comes at times
of a rate Poisson process to check the bulb and will replace the bulb if it is ...
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1answer
83 views
What probability distribution would be suitable for modelling scores in a basketball match?
In football (not American football but what Americans call soccer), it is pretty clear: we may consider the difference of two independent Poisson variables. In basketball, in theory, we could increase ...
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38 views
Probability that epoch of kth arrival in first poisson process lesser than that of jth arrival in second poisson process
Let $S_{1k}$ and $S_{2j}$ denote the epoch of the $k^{th}$ and $j^{th}$ arrival in the first and second poisson counting process, respectively.
The first and second poisson counting process is the ...
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1answer
103 views
Grid based piecewise-stationary Poisson process test
I'm trying to fit a set of data to a variety of Poisson-based models, and have hit a stumbling block when trying to fit a piecewise-stationary Poisson process. What I mean by this is a Poisson process ...
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1answer
334 views
Confidence interval for mean of Poisson distribution witih observations of different length
I am struggling with calculating a confidence interval for my data using a Poisson distribution. The context is manufacturing process quality control, and I think that a Poisson distribution is the ...
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0answers
10 views
Error on a ratio of two measured values each with just a poisson noise estimate
I am trying to estimate the error on some discrete data. I have a total of N counts of which x have the desired characteristic. So x/N is the fraction of the sample I want to plot.
I am assuming ...
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2answers
43 views
idea behind Poisson process property
A property of Poisson process says this:
$N\left ( t \right )$ has independent increments:
if $t_{0}<\cdot \cdot \cdot <t_{n}$
then
$N\left ( t_{1} \right )-N\left ( t_{0} \right ),\...
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16 views
How to model continuous time discrete event rate
I have data consisting of irregular event times and at each event there is either a binary positive or negative result (with approximate probability of a negative result ~5%).
The aim is to assess ...
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384 views
standard error & standard deviation (Poisson counts)
I observe 25 events over a period of 50 seconds. I assume that the events are generated by an underlying Poisson process.
What is the standard deviation of the count?
What is the standard error of ...
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206 views
Difference between standard deviation and standard error (for low count rates)
Similar questions about sample means and counts, but neither directly about count rates.
I get $\lambda=4$ counts over a $T=25$ day interval. If I assume that the counts are yielded by an underlying ...
