# 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.

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### Calibrating a non-homogeneous Poisson process to my data [duplicate]

My question: Let's say I have some data on the cumulative number of infections per day since the start of a pandemic at $t=0$. Since clearly the infection rate changes over time, I want to calibrate a ...
13 views

### What is the optimal measuring time split for limited measuring time between signal+background and background in a Poisson counting experiment?

I’m trying to figure out the best split of time between measuring either background or signal+background in a counting experiment in the case where we have prior estimates for the signal and ...
1 vote
52 views

### Statistical test whether data conforms to a spatial point process--gaza bombing locations

I came across this image on twitter, and it made me think about testing a point process hypothesis. Now this is a politically sensitive image, and I don't want to run afoul of any SE posting ...
1 vote
46 views

### Are these two equivalent forms for the likelihood of a Poisson point process?

I have a Poisson point process in a bounded region $W$. I'm trying to calculate the likelihood of observing a particular set of points within $W$. I'm told that there are two equivalent forms of ...
20 views

### Distribution of a process

I'm having trouble on solving this problem. Given a Poisson process X with parameter lambda and, indipendently, a random variable T with density f(t)=$\theta*e^{(-\theta t)}$, compute the ...
29 views

### Proving that the average number of arrival events as $\lambda t$ given the inter-arrival duration are i.i.d. Exp($\lambda$) random variables

I'm trying to prove a common result for the Poisson process but I'm stuck. Given $T_i$ are i.i.d. $Exp(\lambda)$ random variables (where $\lambda$ is the rate) that represent the duration of arrival ...
87 views

### Let $N(t)$ is a poisson process with rate $\lambda$, $T^* \sim \operatorname{Exp}(\lambda^*)$, find the expectation of $N(\min(t, T^*))$

Currently, my approach is to split $N(\min(t, T^*))$ like the following by the law of total expectation. \begin{align*} &E(N(\min(t,T^*))) = E(N(t \wedge T^*)) \\ = {}&E(N(t\wedge T^*) \...
1 vote
47 views

### Poisson process for small sample size

Suppose I have a data set that contains insurance claims $X$. Each claim also is assigned the calendar year during which the insured event happened. Now, I want to build a frequency/severity model but ...
1 vote
74 views

### Simulating Nonhomogeneous Poisson Process - Conditional distribution of arrival times

For a Poisson process having rate $\lambda$. Given the number of events by time $T$ the set of event times are iid Uniform $(0,T)$ random variables. Suppose that each event are independently counted ...
45 views

### Tail of the maximum of a time-varying Poisson-GP marked process

Consider a time-varying version of the Poisson-GP marked process on the real line as commonly used in Peak Over Threshold (POT) modelling of a variable $Y$. More precisely we have a given time-varying ...
1 vote
68 views

### Let $N(t)$ be a Poisson process, compute $P\{N(s)=1,N(t)=2\}$ for any $0\leq s<t$

I got the answer as $\lambda^2e^{-\lambda t}s(t-s)$ using the properties like independent increment and stationary increments. But I can't seem to understand the steps in the solution of the book. For ...
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I would like to know the meaning or signification of the parameter $\beta$ in this Bayesian model. I have a Poisson model : $s_{i} \mid \lambda_{i} \sim Poisson(\lambda_{i}t_{i})$ Where $\lambda_i\... 0 votes 0 answers 10 views ### Multiple mean comparison test for a Non homogeneous Poisson Process I do statistics for a restaurant in which people arrive at random times and end up paying a random amount of money when they are finished. I am using a Compound Poisson Process to model this, but I ... 1 vote 0 answers 68 views ### Average time in which a product random variable becomes zero Im looking for the optimal time in which a process should be cancelled before it results on financial losses. Say M_n=X_n*Y_n-c(n) for for n =1 to 12 which is the number of hours the process gets ... 2 votes 0 answers 29 views ### A hard core spatial point process that a vehicle encounters as Poisson arrivals? Does there exist a point process$X$in the plane with the following two properties?$X$is hard core. Discs of radius$h$can be centered on the points in$X$without overlapping.$X$is ... 0 votes 1 answer 219 views ### How to appropriately compare colony-forming units (CFUs)? In microbial research, a common way to check growth rates of bacteria is by performing a dilution of the bacterial population and then plating the resulting dilution on a petri dish. After some time, ... 0 votes 0 answers 37 views ### Distribution for a sequence of events which the first one follows a Poisson distribution Imagine a simple website where a user can access and can click on a button that will refresh the page. On average, 50 000 requests are made to the webserver of this website in a month. We can assume ... 2 votes 1 answer 81 views ### Why can weekends cause harmony? The following plot shows power spectra (periodograms) of a sample from$X_t \sim \operatorname{Poisson}(1)$along with that same sample where: Weekends were set to zero Sundays were set to zero ... 0 votes 0 answers 5 views ### Main event time prediction based on different sub events As the title says, I want to predict the time (with a wide error range) of a main event’s first occurrence based on previous sub events that are vary in importance. These previous ‘predictor’ events ... 0 votes 1 answer 257 views ### How to prove the Poisson link function is a canonical link function? So I'm a 3rd year undergraduate doing my thesisin football score models right now. In my thesis I want to include a proof of what the link function for the Poisson distribution is and why it relates ... 1 vote 0 answers 34 views ### Radioactive decay as a Poisson process [duplicate] I am trying to grapple with the following question as I self-study from the chapter on probability in my quantum mechanics textbook (Ballentine's Quantum Mechanics: A Modern Development). ... 2 votes 1 answer 125 views ### Poisson Process: Probability distribution to describe time (distance) to successful event? Given some length of time$t$with successful events occurring in this interval at rate$\lambda$. Assume that only one successful event occurs during this interval of length$t$. Which distribution ... 1 vote 1 answer 77 views ### Count process with standard deviation proportional to its mean What is (is there) the count process, which has its standard deviation proportional to its mean? Note that I am not talking here about Poisson process, which has its variance proportional to mean. ... 0 votes 0 answers 34 views ### Conditional probability for Poisson process Give this question: For a Poisson process with rate λ, find P(N(s) = k|N(t) = n) when s > t. What is the difference if it was given that s < t? If s > t, do the two events become ... 2 votes 1 answer 104 views ### Likelihood for a log Gaussian Cox process (LGCP) Suppose I have a log Gaussian Cox process (LGCP)$X$with log intensity function$\lambda(x)=S(x)$where$S$follows a Gaussian process. Since LGCP still falls under the umbrella of inhomogeneous ... 0 votes 0 answers 31 views ### Poisson process vs selecting based on a probability I want to write a function that returns an error with a rate of n% . I am confused if the following two ways are the same or what is the difference. poisson ... 2 votes 0 answers 22 views ### Aikaike Information Criterion for model with fixed parameter I am trying to fit an inhomogenious, reinforced Poisson process to time series data using maximum likelihood estimation. The inhomogenious rate is $$\lambda = \alpha \cdot f(t,\theta) \cdot (m +n).$$ ... 0 votes 0 answers 44 views ### Parameter estimation of a 'parametric' Poisson process I observe a sample of the form$S = [S_1, S_2, \ldots, S_N]=[(t_1, x_1), (t_2, x_2), \ldots, (t_N, x_N)]$where each$t_i$is an arrival time and$x_i$is the amount of money spent by the$i$th ... 0 votes 0 answers 46 views ### I want to know if age and residual life time of the Poisson process are independent 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$random variable to be$A_t := t-T_{N(t)} $,the$...
If a customer arrives according to a Possion process with rate $\lambda$, how can I show that the time interval $X$ taken to receive $k$ customers is an Erlang-$k$ random variable with parameters $n$ ...