Questions tagged [poisson-process]
For questions about the theory or applications of the Poisson process, one of the most widely applied point processes in statistics and elsewhere.
258
questions
0
votes
0answers
16 views
Multivariate Bayesian Car Model Result
I have developed a multivariate Bayesian Car model for three crash severity level analysis. I found that the covariance for both heterogenous effects and the spatial effect is not significant for any ...
4
votes
1answer
56 views
Confidence interval over multiple Poisson distributions
There are five different periods (one following each other) of unequal lengths where one specific event is happening such that it follows Poisson distributions (e.g. meteorite falls). It is always the ...
0
votes
0answers
12 views
Computing a Probability of Poisson Stochastic Process
NOTE: The questions can answered independently, but it is preferable to follow through the reasoning.
Suppose $\{N_t\}$ is a Poisson process with intensity $\lambda=4$. I want to compute
$$\mathbb{P}...
4
votes
1answer
362 views
Probability of a 500 year flood occuring in the next 100 years - comparison of approaches
I'm looking at this problem
A $500$-year flood is one that occurs once in every $500$ years.
a) What is the probability of having at least $3$ such floods in $500$ years?
b) What is the ...
0
votes
0answers
16 views
Poisson process as a spatial process
Let's consider a Poisson process on the line with rate parameter $\lambda$. There are two ways to think about this:
In any interval $[a,b)$ the expected number of events is distributed as a Poisson ...
0
votes
0answers
9 views
Simultaneous Poisson processes
The arrival of taxis at a taxi stand is Poisson at rate $\lambda$ per hour. The arrival of people to the stand is also Poisson but at rate $\mu$ per hour.
Taxis do not stop (they leave empty) if ...
0
votes
0answers
59 views
Pareto/NBD with time-varying covariates
I am trying to incorporate time-varying covariates into the Pareto/NBD model (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.597.3165&rep=rep1&type=pdf) Model assumptions start at ...
1
vote
0answers
22 views
Does it make sense to infer a rate (as a probability distribution or upper limits) for a Poisson process if there are “no events”
I have an inhomogeneous Poisson process with a rate $\lambda (\mathbf{t})$ defined on some parameters $\mathbf{t}$. I am trying to infer $\lambda (\mathbf{t})$ from some data, which are events (really ...
2
votes
1answer
32 views
Smallest gap between $n$ Poisson events
Given a Poisson process with parameter $\lambda,$ run time forward until the first $n>1$ events occur. What is the distribution of the smallest time interval between two events?
This is surely ...
0
votes
0answers
13 views
MLE estimation of poisson intensity for a normal compounded poisson process
I was working on a project where a time series is modeled as compound poisson distribution with normally distributed jump levels and I want to get MLE estimator for the poisson intensity, jump mean, ...
0
votes
1answer
26 views
Poisson regression for count data that is not Poisson distributed?
Is it true that Poisson regression is used to model count data? But not all count data follow a Poisson distribution? Then you can still use Poisson regression in that case?
0
votes
0answers
37 views
Comparison of Poisson cumulative probabilities between models with different rates
I'm building an anomaly detection model using the Poisson distribution.
I'm calculating the Cumulative Probability: P(X >= x) for thousands of count data, all of them with different rates.
I need to ...
2
votes
1answer
47 views
Poisson Fun exercise question
Based on your understanding of the Poisson process, determine the numerical values of $a$ and $b$ in the following expression.
$$ \int_{t}^∞ {λ^6τ^5e^{−λτ} \over 5!}dτ= \sum_{k = a}^b{(λt)^ke^{-...
0
votes
1answer
16 views
Which distribution or process should be used for wearout reliability modeling?
When modeling the reliability of a system, it is usual to use exponential distribution to model errors that occur randomly throughout the system's useful lifetime (the middle part of the well-known ...
0
votes
1answer
25 views
Extension of Poisson Parameter for Different Temporal Interval
Suppose $Y$ follows a Poisson distribution with parameter $ \lambda $ that explains a temporal Poisson process over an interval of 30 seconds.
Now, it stands to reason that for an interval of 60 ...
1
vote
1answer
14 views
Test and CI for unstable Poisson process
I have empirical data on a process that I assume is Poisson with a given mean, say $\mu$ (unknown). The data is of the form $(x_i, 1\leq i\leq n)$ for $n$ consecutive time periods.
I am concerned ...
1
vote
2answers
57 views
Upper confidence bound for Poisson process rate parameter
I am interested in computing an upper confidence bound for the rate parameter, $\lambda$, in a Poisson process. Specifically, I have a set of observations $$
X_\text{obs} = \{(n_1,t_1), \ldots, (n_N,...
3
votes
2answers
87 views
Alternating between two states {A, B} each with exp distributed durations. What's the probability of state=A at time t?
Say I have a light bulb that can be on (A) or off (B). It alternates between being state A or B. It will be in state A for a duration a ~ exp(α), and in state B for duration b ~ exp(β), (...
2
votes
2answers
141 views
Conditional distribution of arrival times in Poisson process
Suppose I know over a window $[0, T)$ that I have observed $n$ samples from a poisson process $N_t \sim p(n|\lambda t) = \frac{1}{n!}(\lambda t)^{n}\exp(-\lambda t)$.
What is the conditional ...
0
votes
0answers
20 views
Poisson Process With Two Different Rates
I am looking to investigate a Poisson Process that has two separate rates; essentially, there is a season of games that each have their own points rate (per minute). Comprehensively, over the season, ...
0
votes
0answers
16 views
Overdispersion under different longitudinal constraints
I have panel data (police districts observed across many months) and I am modeling a count outcome. Fitting a Poisson model (simple pre/post comparison with 6 months of data) results in significantly ...
1
vote
0answers
20 views
How to compute the expected number of events in the following conditional renewal process?
I have a stochastic point process with event times $\{x_1, x_2, ...\} $ and I want to compute the expected number of events $n(T)$ over the interval $[0,T]$. The point process is generated as follows:
...
0
votes
0answers
22 views
How to show the inter-arrival time variance of a Cox process driven by a stationary Poisson process of constant intensity $\lambda$ is $3\lambda$
Ideas on how to show that the variance of a doubly-stochastic Poisson process(aka a Cox process) driven by a homogeneous(stationary) Poisson process of intensity $\lambda$ is $3\lambda$ ?
I've come ...
1
vote
0answers
29 views
To detect which of the three processes more-aptly fits Poisson process
There happens to be a type of generic poll, which can be described as:-
A participant can vote only once and for only 1 of 3 running candidates - Red, Blue and Green. A random voter starts the ...
0
votes
0answers
21 views
Solving an equation with logarithms of negative numbers
I'm modelling a bunch of non-homogeneous poisson processes. I'm impossing on them a linear-log functional form. you end up with two parameters, theta1 and theta0 for each estimated process.
The ...
2
votes
0answers
22 views
Understanding the solution to a problem about a homogeneous Poisson process
This is probably easy, but right now I can't figure it out, so bear with me. The question is:
Let $\{N(t),t\ge 0\}$ be a homogeneous Poisson process on $(0,\infty)$ with rate $\lambda$. Let $\{S_i, i=...
2
votes
1answer
70 views
Manually simulating Poisson Process in R
The following problem tells us to generate a Poisson process step by step from $\rho$ (inter-arrival time), and $\tau$ (arrival time).
One of the theoretical results presented in the lectures ...
0
votes
1answer
26 views
Is this problem worked out correctly? [closed]
Calls are received at a company call center according to a Poisson process at the rate of five calls per minute.
(a) Find the probability that no call occurs over a 30-second period.
(b) Find the ...
0
votes
1answer
17 views
Expectation of arrival times
Let $(N_t)_t$ be a Poisson process with parameter λ = 2. By $τ_k$ denote the time of the k-th arrival (k = 1, 2, . . .), and by $ρ_k = τ_k −τ_{k−1}$ - the interarrival time between the (k−1)th and kth ...
1
vote
1answer
47 views
How to check if a data is poisson sampled?
I was reading one article which develops a theory for the Poisson sampled data. That is the data is collected over time-points $\{T_k, k>1\}$, which are jump-moments of a homogeneous Poisson ...
0
votes
1answer
56 views
What is the difference between time, arrival-time, and inter-arrival-time is poisson process?
Let $(N_t)_t$ be a Poisson process with parameter λ = 2. By $τ_k$ denote the time of the k-th arrival (k = 1, 2, . . .), and by $ρ_k = τ_k −τ_{k−1}$ - the interarrival time between the (k−1)th and kth ...
0
votes
0answers
22 views
How to properly truncate exponential distribution to represent random memory-less arrivals?
Typically, Poisson and exponential distributions are used to represent random memory-less arrival processes. It has come to my attention, however, that a more realistic distribution is a truncated (or ...
0
votes
0answers
27 views
Distribution of the y-coordinate of a 2D Poisson Process
Consider a Poisson process with rate 1 over $R_+ \times R$. I found this term in a scientific article, I assume it means :
-The number of points in a subset $A$ of $R_+ \times R$ has the Poisson ...
0
votes
1answer
28 views
Generating Arrival Counts Given Time
I'm looking to generate arrival counts given a length of time for a Poisson process. This is similar to another question, however rather than generate the particular arrivals, I'd like to just get the ...
0
votes
1answer
49 views
Poisson process - measuring empirical changes
I was recently discussing Poisson processes in industrial settings with a colleague and he came up with a great question.
Let's say that defective products on a manufacturing line tend to occur ...
0
votes
1answer
66 views
Simulating a (discretized) Cox process via binomial sampling
Let X be a Cox process (doubly-stochastic Poisson process) driven by a Poisson process with fixed intensity(rate) $\lambda=50$ , and choose some small time interval $dt=0.01$ . Is the proper way to ...
0
votes
0answers
34 views
Inhomogeneous K-function to indicate need for spatial dependence/interaction term in Poisson point process model
I am mapping and modelling a disease of sheep. I have approx 4200 point locations in my dataset, each of which represents the centroid of a given sheep farm.
I have created a K-function difference ...
1
vote
1answer
75 views
Proof for simulation of NHPP by thinning
Background:
I'm trying to show equivalency between the density function for a non-homogenous exponential process (NHEP?), (i.e. the arrival times of events generated by a non-homogenous Poisson ...
3
votes
0answers
50 views
two independent Poisson Arrivals
I have two types of customers (type 1 and type 2) enter a shop. Their arrival processes are independent and follow Poisson process with the arrival rates of $\lambda_1$ and $\lambda_2.$
Consider two ...
1
vote
0answers
33 views
Expected time to wait for no events to occur within a sliding window assuming Poissson process
I wish to model the following:
I am maintaining a sliding window (history) of 10 samples of the output of a signal detector.
I model the probability of a detection failure (i.e absence of signal) as ...
0
votes
0answers
22 views
Poisson Process Simple Question
The number of customers that arrive at a cashpoint in an hour is distributed poisson($\lambda$). Suppose that each arriving customer makes a draft. Let $Y_i$ denote the amount of money $i^{th}$ ...
1
vote
1answer
294 views
Probability Density function of Poisson distribution
This is an assignment I got for my course on Stochastic Processes:
Let us consider a random variable X distributed as a Poisson P (λ)
where λ ∼ [0.5, 1].
(a) Which are the unconditional ...
1
vote
1answer
590 views
Poisson and conditional probability
Admit that the number of participants who intend to enroll in a given training follows a Poisson distribution with a mean of $12.$ If there is not a minimum of five enrollments, training is not ...
0
votes
1answer
23 views
Compound risk poisson models
I was just working through this question.
A compound Poisson risk model is used to model the total claims S
experienced by an insurance company over one year, of the form:
$S = X_1 + ... + X_n$
...
0
votes
1answer
67 views
Intensity function in Poisson random effect model
I have a somewhat general question about intensity functions in Poisson random effect models.
Consider the Poisson random effects model in which conditional on a random effect $u$, an individual ...
0
votes
0answers
12 views
Coverage of a Boolean model
The following problem I am working on involves the coverage of Boolean Model (BM).
The grains in the BM are discs with fixed radius $r$. $A$ is the set of this BM in space $ \mathbb R^2 $ and $K$ ...
0
votes
1answer
70 views
Correcting data using poisson-regression
I'm new to stats and
I was wondering if anyone had any good resources that could explain to me:
How one can correct their data (false-positives) using Poisson-regression.
I've been looking for some ...
0
votes
1answer
136 views
How to model arrival times in discrete event simulation where arrivals vary with time of day and day of week
I'm looking for suggestions on how to model inter-arrival times in a discrete event simulation where the arrival rate is highly dependent on the day of week and the hour of day.
For example:
A ...
2
votes
1answer
59 views
The probability distribution of waiting time until two exponentially distributed events with different parameters both occur
I am working on a problem related to the waiting time until a parking garage is empty. We are given that the cars independently spend an
exponential distributed time in the parking garage, with ...
1
vote
0answers
89 views
Partial imputation of missing dates
I'm working with dataframes (one for each of 185 locations) that shows sums of occurrences for each calendar date. There are no 0 values for occurrences in the entire dataset. There are several ...
