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.
0
votes
0answers
10 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
11 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
25 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 ...
1
vote
0answers
20 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
47 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
16 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
29 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
45 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
16 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
24 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
19 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
43 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
0answers
14 views
Simulating a (discretized) Cox process via binomial sampling
Let X be a Cox process (doubly-stochastic Poisson process) with fixed intensity(rate) $\lambda=50$ , and choose some small time interval $dt=0.01$ . Is the proper way to simulate this, by letting Y ...
0
votes
0answers
25 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
50 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
42 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
31 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
21 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
203 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
90 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
22 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
51 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
8 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
67 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
71 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
47 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
54 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 ...
0
votes
1answer
17 views
Computing probabilities of comparisons of exponential random variables
I am working on a question for class where there are two patients waiting on kidneys. The arrival of transplant kidneys is a Poisson process with rate λ. A will die after an exponential time with rate ...
0
votes
0answers
31 views
question regarding a stochastic process
Say let $Y_1, Y_2,...$ be a family of i.i.d random variables with a common $\mu$ and common variance $\sigma^2$. Let $N_t$ be a Poisson process with rate $\lambda >0 $. Assume that $N_t$ is ...
0
votes
0answers
21 views
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 ...
2
votes
0answers
320 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 ...
0
votes
0answers
40 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 ...
2
votes
1answer
1k 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 ...
1
vote
1answer
101 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 ...
2
votes
1answer
204 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, ..., ...
2
votes
1answer
382 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:
...
0
votes
2answers
623 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:
...
0
votes
1answer
50 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 ...
2
votes
0answers
18 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 ...
1
vote
1answer
1k 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 ...
0
votes
1answer
31 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'...
0
votes
0answers
48 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/...
0
votes
1answer
39 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 ...
0
votes
1answer
68 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 $\frac34$.
...
1
vote
1answer
25 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 ...
1
vote
0answers
92 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 ...
0
votes
1answer
299 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 ...
1
vote
0answers
16 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 ...
