What is Poisson process: Definition and 54 Discussions

In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space. The Poisson point process is often called simply the Poisson process, but it is also called a Poisson random measure, Poisson random point field or Poisson point field. This point process has convenient mathematical properties, which has led to it being frequently defined in Euclidean space and used as a mathematical model for seemingly random processes in numerous disciplines such as astronomy, biology, ecology, geology, seismology, physics, economics, image processing, and telecommunications.The process is named after French mathematician Siméon Denis Poisson despite Poisson never having studied the process. Its name derives from the fact that if a collection of random points in some space forms a Poisson process, then the number of points in a region of finite size is a random variable with a Poisson distribution. The process was discovered independently and repeatedly in several settings, including experiments on radioactive decay, telephone call arrivals and insurance mathematics.The Poisson point process is often defined on the real line, where it can be considered as a stochastic process. In this setting, it is used, for example, in queueing theory to model random events, such as the arrival of customers at a store, phone calls at an exchange or occurrence of earthquakes, distributed in time. In the plane, the point process, also known as a spatial Poisson process, can represent the locations of scattered objects such as transmitters in a wireless network, particles colliding into a detector, or trees in a forest. In this setting, the process is often used in mathematical models and in the related fields of spatial point processes, stochastic geometry, spatial statistics and continuum percolation theory. The Poisson point process can be defined on more abstract spaces. Beyond applications, the Poisson point process is an object of mathematical study in its own right. In all settings, the Poisson point process has the property that each point is stochastically independent to all the other points in the process, which is why it is sometimes called a purely or completely random process. Despite its wide use as a stochastic model of phenomena representable as points, the inherent nature of the process implies that it does not adequately describe phenomena where there is sufficiently strong interaction between the points. This has inspired the proposal of other point processes, some of which are constructed with the Poisson point process, that seek to capture such interaction.The point process depends on a single mathematical object, which, depending on the context, may be a constant, a locally integrable function or, in more general settings, a Radon measure. In the first case, the constant, known as the rate or intensity, is the average density of the points in the Poisson process located in some region of space. The resulting point process is called a homogeneous or stationary Poisson point process. In the second case, the point process is called an inhomogeneous or nonhomogeneous Poisson point process, and the average density of points depend on the location of the underlying space of the Poisson point process. The word point is often omitted, but there are other Poisson processes of objects, which, instead of points, consist of more complicated mathematical objects such as lines and polygons, and such processes can be based on the Poisson point process. Both, the homogeneous Poisson point process and the nonhomogeneous Poisson point process are particular cases of the generalized renewal process.

View More On Wikipedia.org
  1. SunilS

    A Relationship between Poisson distribution and Poisson Process

    Apologies if this has been discussed elsewhere. I know a Poisson process implies a Poisson distribution, but does a Poisson distribution imply a Poisson process? and does the absence of a Poisson distribution imply the absence of a Poisson process? TIA - Sunil
  2. user366312

    I Difference between Time, Arrival-Time & Inter-Arrival-Time in Poisson Process

    . The above are some of the typical problems related to Poisson Process. I need to understand the difference between time, inter-arrival time, and arrival time in this regard. Say, we start our counting from 9:00 AM and count up to 10:00 AM. Image-1: arrival process. 1. 1st call comes at...
  3. Mehmood_Yasir

    Pedestrian at a road crossing

    Homework Statement Pedestrians approach to a signal for road crossing in a Poisson manner with arrival rate ##\lambda## per sec. The first pedestrian arriving the signal pushes the button to start time ##T##, and thus we assume his arrival time is ##t=0##, and he always see ##T## wait time. A...
  4. Mehmood_Yasir

    Probability and pedestrian wait time density function

    Homework Statement Pedestrians approach to a signal at the crossing in a Poisson manner with arrival rate ##\lambda## arrivals per minute. The first pedestrian arriving the signal starts a timer ##T## then waits for time ##T##. A light is flashed after time T, and all waiting pedestrians who...
  5. Livio Arshavin Leiva

    A What exactly is a "rare event"? (Poisson point process)

    These days I've been reading in the internet about the Poisson Distribution because that was a concept I couldn't manage to understand completely when I studied it, so since then I've been always quite curious about Poisson processes, and how there are a lot of natural phenomena (mostly the...
  6. Mehmood_Yasir

    I Conditional Expectation Value of Poisson Arrival in Fixed T

    Assume a Poisson process with rate ##\lambda##. Let ##T_{1}##,##T_{2}##,##T_{3}##,... be the time until the ##1^{st}, 2^{nd}, 3^{rd}##,...(so on) arrivals following exponential distribution. If I consider the fixed time interval ##[0-T]##, what is the expectation value of the arrival time...
  7. F

    Optimal Stopping Strategy for Winning Game with Two Bells

    Homework Statement You are playing a game with two bells. Bell A rings according to a homogeneous poisson process at a rate r per hour and Bell B rings once at a time T that is uniformly distributed from 0 to 1 hr (inclusive). You get $1 each time A rings and can quit anytime but if B rings...
  8. mertcan

    I N events occur in one Poisson process before m events

    guys, I have a very ımportant question. First let me introduce parameters: $$S^A_1 = \text{first arrival of A event}, and S^B_1= \text{first arrival of B event}, and S^C_1=\text{ first arrival of C event}$$, then probability of $$P(S^A_1<S^B_1) = \frac {\lambda_A} {\{ \lambda_A + \lambda_B \}...
  9. CynicusRex

    I Poisson process approximation error

    X = # of cars that pass in one hour E(X) = λ = n * p λ cars/1hour = 60min/hour * (λ/60) cars/min In this old video (5:09) on poisson process Sal asks: "What if more than one car passes in a minute?" "We call it a success if one car passes in one minute, but even if 5 cars pass, it counts as 1...
  10. J

    Autocorrelation function of a Wiener process & Poisson process

    Homework Statement 3. The Attempt at a Solution [/B] ***************************************** Can anyone possibly explain step 3 and 4 in this solution?
  11. J

    Understanding the Characteristic Function in Solving Poisson Process Problems

    Homework Statement Insects land in the soup in the manner of a Poisson process with intensity lambda. Insects are green with probability p, independent of the color of the other insects. Show that the arrival of green insects is a Poisson process with intensity p*lambda. Homework Equations3...
  12. S

    A question about Poisson process (waiting online)

    Hey guys, I encounter a question (maybe a silly one )that puzzles me. Nt is a Poisson process and λ is the jump intensity.Since the quadratic variation of Poisson process is [N,N]t=Nt, and Nt2-[N,N]t is a martingale, it follows that E[Nt2]=E[[N,N]t]=λ*t. On the other hand, the direct calculation...
  13. H

    About stochastic process....Help please

    Given a Gaussian process X(t), identify which of the following , if any, are gaussian processes. (a)X(2t) solution said that X(2t) is not gaussian process, since and similarly Given Poisson process X(t) (a) X(2t) soultion said that X(2t) is not poisson process, since same reason above...
  14. L

    Poisson process and exponential distribution arrival times

    Homework Statement Customers arrive in single server queue to be serviced according to Poisson process with intensity 5 customers an hour. (a) If the customers begin to arrive at 8am, find the probability that at least 4 customers arrived between 9am and 10am. (b) Find the probability that the...
  15. I

    MHB 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 $[t, t+\Delta t]$, $P\left \{ k \;\text{events in}\; [t, t+\Delta t] \right \}=\frac{\text{exp}(-\lambda \Delta t)(\lambda \Delta t)^k}{k!}$. And therefore, event count in...
  16. I

    MHB Proof about an Inhomogeneous Poisson Process

    We know that an inhomogeneous Poisson process is a process with a rate function $\lambda(t)$. That is, for any time interval $[t, t+\Delta t]$, $P\left \{ k \;\text{events in}\; [t, t+\Delta t] \right \}=\frac{\text{exp}(-s)s^k}{k!}$, where $s=\int_{t}^{t+\Delta t}\lambda(t)dt$. And Here is the...
  17. P

    Probability distribution of first arrival time in Poisson Process

    According to wiki: http://en.wikipedia.org/wiki/Poisson_process The probability for the waiting time to observe first arrival in a Poisson process P(T1>t)=exp(-lambda*t) But what is the Probability Distribution P(T1=t) of the waiting time itself? How to calculate that?
  18. U

    MHB Poisson Process - Number of cars that a petrol station can service

    Question: A single-pump petrol station is running low on petrol. The total volume of petrol remaining for sale is 100 litres. Suppose cars arrive to the station according to a Poisson process with rate \lambda, and that each car fills independently of all other cars and of the arrival...
  19. S

    Poisson Process Probability Question

    Hello, I have this one problem but have no idea how to get started. Avg. number of accidents is .4 accidents / day (Poisson Process) What is the probability that the time from now to the next accident will be more than 3 days? What is the probability that the the time from now to...
  20. B

    Poisson process, question about the definition.

    Hi, I have a question about the definition of the poisson process. Check out the definition here: Would you say that one can prove point (2) from point (3)? The reason I have some discomfort about this is that something seems to be hidden in the poisson distribution to make it all work? For...
  21. P

    Simulating random process (poisson process)

    Homework Statement I have a physical system, which I know the time average statistics. Its probability of being in state 1 is P1, state 2:P2 and state 3:P3. I want to simulate the time behavior of the system.Homework Equations N/AThe Attempt at a Solution I assume the rate of transition event...
  22. C

    Calculating Probability of a Poisson Process w/ Parameter λ

    I need some help on the following question: Let N() be a poisson process with parameter \lambda . I need to find that probability that N((1,2]) = 3 given N((1,3]) > 3 I know that this is equal to the probability that P(A \cap B) / P(B) where A = N((1,2]) and B = N((1,3]) >...
  23. beyondlight

    How to solve poisson process probabilities

    Homework Statement Let X(t) and Y(t) be independent Poisson processes, both with rates. Define Z(t)=X(t)+Y(t). Find E[X(1)|Z(2)=2]. 2. The attempt at a solution...
  24. H

    MHB Probability that all N_Q packets arrived in [0,t], in a Poisson process

    Arrivals are Poisson distributed with parameter \lambda. Consider a system, where at the time of arrival of a tagged packet, it sees N_Q packets. Given that the tagged packet arrives at an instant t, which is uniform in [0, T], what is the probability that all N_Q packets arrived in [0,t]?This...
  25. J

    Poisson Process Conditional Distribution

    Homework Statement X_t and Y_t are poisson processes with rates a and b n = 1,2,3...Find the CDF F_X{}_t{}_|{}_X{}_t{}_+{}_Y{}_t{}_={}_n(x)Homework Equations The Attempt at a Solution F_X{}_t{}_|{}_X{}_t{}_+{}_Y{}_t{}_={}_n(x) =P(X_t<x|X_t+Y_t=n) =\frac{P(X_t<x,X_t+Y_t=n)}{P(X_t+Y_t=n)} Not...
  26. S

    Poisson Process: interevent times

    Homework Statement Consider a one-way road where the cars form a PP(lambda) with rate lambda cars/sec. The road is x feet wide. A pedestrian, who walks at a speed of u feet/sec, will cross the road if and only if she is certain that no cars will cross the pedestrian crossing while she is on...
  27. P

    MHB Poisson process derivation

    Let customers arrive according to a poisson process with parameter st and let $X_{t}$ denote number of customers in the system by time t. Consider an interval [t,t+h] with h small. Show that P(1 arrival)= sh + o[h], P(more than one arrival)=o[h] and P(no arrival)=1-sh+o[h]. I know P(1...
  28. S

    Stochastic modelling, poisson process

    Homework Statement Suppose a book of 600 pages contains a total of 240 typographical errors. Develop a poisson approximation for the probability that three partiular successive pages are error-free. The Attempt at a Solution I say that the number of errors is poissondistributed...
  29. G

    Problem related to the compound Poisson process (?)

    Dear all, I wonder if anyone has come across this problem before and could point me to a relevant ref or tell me what terms I might search for: I am interested in a continuous time process in which two alternating events (call them A and B) occur. Each event has an exponentially...
  30. J

    Poisson Process and Stress Fractures in Railway Lines

    Homework Statement Suppose that stress fractures appear in railway lines according to a Poisson process at a rate of 2 per month. a)Find the probability that the 4th stress fracture on the railway line occurred 3 months after the process of checking the new railway lines. b)Suppose new...
  31. S

    Stochastic Process, Poisson Process

    Hi, I need some help with this hw 1. Suppose that the passengers of a bus line arrive according to a Poisson process Nt with a rate of λ = 1 / 4 per minute. A bus left at time t = 0 while waiting passengers. Let T be the arrival time of the next bus. Then the number of passengers who...
  32. T

    Newton-Raphson method in non-homogeneous poisson process

    Homework Statement The rate of occurrence of events in a non-homogeneous Poisson process is given by: λ(t)=12t e-2t. (c) Find the p.d.f. of the time until the first event occurs after time t = 1. (e) After what time is it 95% certain that no further events will occur? Homework Equations...
  33. A

    Calculating Probability in Poisson Process Problem | Z(t-c)=m, Z(t)=k

    Given a poisson process Z(t) with a given rate lamda, k and m nonnegative integers and t and c real and positive numbers, calculate the probability: P(Z(t-c)=m | Z(t)=k) thanks
  34. M

    Characteristic Function of a Compound Poisson Process

    Hello, I am trying to find a characteristic function (CF) of a Compound Poisson Process (CPP) and I am stuck :(. I have a CPP defined as X(t) = SIGMA[from j=1 to Nt]{Yj}. Yj's are independent and are Normally distributed. So, in trying to find the CF of X I do the following: (Notation...
  35. F

    Independence in Poisson Process

    I'm studying the Poisson Process (rate R) and I'm hung up on the issue of dependence. This seems like and easy question but I have no background in probability whatsoever. By definition, the number of events in disjunction time intervals are independent. Okay. Fine. But say we have an...
  36. K

    The quadratic covariation of Brownian motion and poisson process

    Hi: I want to know the quadratic covariation of Brownian motion B(t) and poisson process N(t).Is it B(t)? Thanks !
  37. Saladsamurai

    Probability Poisson Process and Gamma Distribution

    Homework Statement The Attempt at a Solution Part (a) is no problem, it is simply P(X>10) = 1 - P(X<=10) which requires the use of tabulated cumulative poisson values. Part (b) is throwing for a loop. I know that I need to invoke the Gamma distribution since that is what the...
  38. A

    Poisson Process Conditonal Probabilities

    Hey I'm really struggling with this: What is the expected value of a poisson process (rate λ, time t) given that at least one even has occured? I was told the best way was to find the conditional distribution first. So this is: P(Xt=z | Xt≥1) = P(Xt=z, Xt≥1) / (PXt≥1) = P(Xt=z) /...
  39. T

    Poisson process with different arrival rates

    Homework Statement I cannot figure out this example: suppose that initially individuals enter a room from one door according to a Poisson process with arrival rate lambda1. Suppose that as soon as one inidividual enters, this door is shut down and a second door is open. The numer of...
  40. D

    Stochastic Processes, Poisson Process | Expected value of a sum of functions.

    Homework Statement Suppose that passengers arrive at a train terminal according to a poisson process with rate "$". The train dispatches at a time t. Find the expected sum of the waiting times of all those that enter the train. Homework Equations F[X(t+s)-X(s)=n]=((($t)^n)/n!)e^(-$t))...
  41. R

    What is the Error in My Approach to the Poisson Process Problem?

    I'm a bit frustrated with this one... Let (X_t)_{t\geq 0} be a Poisson Process with rate \lambda Each time an 'arrival' happens, a counter detects the arrival with probability p and misses it with probability 1-p. What is the distribution of time, T until the first particle is detected? I...
  42. D

    What is the Probability of 2 Events Occurring in a Poisson Process?

    Homework Statement Events X, Y, Z are all Poisson processes. Event X has a rate of 1 per unit time , event Y has a rate of 2 per unit time and event Z has a rate of 3 per unit time. Find the probability that 2 events (of any type) occur during the interval (0, 3). Homework Equations...
  43. R

    Probability of 1st Arrival From Poisson Process of Rate $\lambda$

    I did this question, but I'm unsure of my reasons behind it. I was hoping someone here could go through the problem for me. I got the answer 1/\lambda - 1/(\lambda + \mu). I did so by integrating, \int_0^\infty P(\text{one event from } \lambda \text{ in }(0, t]) \times P(\text{zero event...
  44. R

    Quadratic Variation of a Poisson Process?

    Hey guys, This is my first post on PhysicsForums; my friend said that this was the best place to ask questions about math. Anyways, I have to find the Quadratic Variation of a Poisson Process. My professor doesn't have a class textbook (just some notes that he's found online), and...
  45. K

    Poisson process: compute E[N(3) |N(2),N(1)]

    note: N(t) is the number of points in [0,t] and N(t1,t2] is the number of points in (t1,t2]. Let {N(t): t≥0} be a Poisson process of rate 1. Evaluate E[N(3) |N(2),N(1)]. If the question were E[N(3) |N(2)], then I have some idea... E[N(3) |N(2)] =E[N(2)+N(2,3] |N(2)] =E[N(2)|N(2)] +...
  46. T

    Stochastic Processes - Poisson Process question

    I had this problem on my last midterm and received no credit for these parts. 1. Express trains arrive at Hiawatha station according to a Poisson process at rate 4 per hour, and independent of this, Downtown local buses arrive according to a Poisson process at rate 8 per hour. a. Given that 10...
  47. Q

    Question about probability and poisson process

    Hi all, I have a question about probability. Can you help me? There are 2 events: - Customer A arrives the system B in accordance with a Poisson process with rate Lambda1 - Customer A arrives the system C in accordance with a Poisson process with rate Lambda2. Given that Poisson...
  48. Q

    A problem related to Poisson process

    Hi all, I have a probability problem. Can you help me? Thank you! Here is the problem: Consider the queueing system, there are n customers 1, 2, ...N. Customer 1 arrives in accordance with a Poisson process with rate Lamda, customer 2 arrives in accordance with a Poisson process with rate...
  49. B

    Poisson Process Homework: Chance of Mushrooms in One Yard

    Homework Statement If you find a mushroom, what is the chance that at least one more will be within one yard from it ? What is the chance that there is exactly one mushroom within the distance one yard from the point you stay? The mushrooms grow in a forest randomly , with density 0.5...
  50. M

    Poisson Process - Prob Theory

    Homework Statement Cars pass a certain street location according to a Poisson process with rate lambda. A woman who wants to cross the street at that location wait until she can see that no cars will come by in the next T time units. Find the probability that her waiting time is T...