Poisson Definition and 509 Threads

In probability theory and statistics, the Poisson distribution (; French pronunciation: [pwasɔ̃]), named after French mathematician Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.
For instance, a call center receives an average of 180 calls per hour, 24 hours a day. The calls are independent; receiving one does not change the probability of when the next one will arrive. The number of calls received during any minute has a Poisson probability distribution: the most likely numbers are 2 and 3 but 1 and 4 are also likely and there is a small probability of it being as low as zero and a very small probability it could be 10. Another example is the number of decay events that occur from a radioactive source during a defined observation period.

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  1. F

    Complex form of poisson equation

    Hi, I am trying to solve the complex form of poisson equation ∇[ε(x)∇V(x)] = -1/εo ρ(x) with complex permittivity. If I introduce complex permittivity ε = εr - j σ/(εow), then I must introduce a complex potential ,V = Vr + jVi. That means the charge density, ρ, must also take a complex...
  2. T

    How Do You Calculate One Sigma Confidence Intervals for Poisson Events?

    I have been analyzing some data at work, and I have measured the occurrence rates of some event. How do I give a one sigma confidence interval to go along with it, assuming it is a Poisson event? For example, I found that something occurs 20 out of 10 000 times, something else occurs 43 out of...
  3. B

    Green's function for Poisson Equation

    Hi, I am working on finding a solution to Poisson equation through Green's function in both 2D and 3D. For the equation: \nabla^2 D = f, in 3D the solution is: D(\mathbf x) = \frac{1}{4\pi} \int_V \frac{f(\mathbf x')}{|\mathbf x - \mathbf x'|} d\mathbf{x}', and in 2D the solution is: D(\mathbf...
  4. A

    Uniform Field & Poisson equation Mismatch?

    Hi, I'm getting some confusing results and can't figure out what is wrong Suppose we have a uniform field E=[0,0,E_z] in a dielectric media. By E=-\nabla\psi we can deduce that \psi(x,y,z)=-z E_z But, taking the Laplacian \nabla^2\psi=\frac{\partial^2 (-zE_z)}{\partial z^2}=0...
  5. H

    Uniform Convergence of Poisson Kernel on [-π, π] minus (-a, a)

    Homework Statement show that the integral of the poisson kernel (1-r^2)/(1-2rcos(x)+r^2) converges to 0 uniformly in x as r tend to 1 from the left ,on any closed subinterval of [-pi,pi] obtained by deleting a middle open interval (-a,a) Homework Equations the integral of poisson...
  6. E

    Solving Non-Homogeneous Poisson Equation: Techniques and Practice Problems

    Solve: Δu=-1, u(1, theta)=sin(theta). 0<r<1, -pie<theta<pie, u finite at r=0 What I've done: u=u1+u2. Δu2=0, with u(1, theta)=sin(theta). So eventually u2=rsin(theta). The u1 problem however I am not sure how to solve. Eigenfunction expansion doesn't seem like it would work (though not...
  7. N

    Applying Symmetry in Poisson Superfish for Cylindrically Symmetric RF Problems

    Does Poisson Superfish apply median-plane symmetry to cylindrically symmetric rf problems (or can it do so)? When I was introduced to the program, I was told that this symmetry was incorporated, but I've not seen anything in the documentation supporting that assertion. On the other hand, a...
  8. T

    Does the Poisson Bracket Always Equal Zero When Both Observables Start at Zero?

    Hello, If you have two observables f and g both of which start off as: f =0 and g =0 and you evaluate their possion bracket: {f,g}, will it necessarily be equal to zero? Also, if just f=0 and g wasn't zero, would {f,g} =0? Thanks!
  9. A

    Poisson distribution with efficiency problem

    Hi, I have a problem with determining the probability distribution function of the number n of detector counts in a given time t. I am assuming the events follow exponential distribution ε(t,λ) = λexp(-λt). Now if that was everything it would simply be a Poisson distribution, however, what I...
  10. S

    Solving Poisson Equation by using FDM

    I need help from anyone urgently, I need C code for Solving Poisson Equation has known source with Neumann condition by using FDM (finite difference method) in 2D problem.
  11. M

    Why use poisson to model arrival of clients

    Hi Almost every text use as example poisson/exponential distribution to model clients arrival. What makes this distribution so good to fit in these cases? Please math arguments Regards
  12. J

    Poisson brackets for a particle in a magnetic field

    I'm struggling to understand Poisson brackets a little here... excerpt from some notes: I am, however, stumped on how this Poisson bracket has been computed. I presume ra and Aa(r) are my canonical coordinates, and I have \dot{r}_a = p_a - \frac{e}{c}A_a (r) with A_a = \frac{1}{2}\epsilon...
  13. E

    Transition from Poisson bracket into Canonical Commutation Relations

    In book http://www.phy.uct.ac.za/people/horowitz/Teaching/lecturenotes.pdf in section 2 it is described transition from Poisson bracket into Canonical Commutation Relations. But it is written The experimentally observed phenomenon of incompatible measurements suggests that position and...
  14. P

    MHB How Does the Poisson Process Model Customer Arrivals Over Time?

    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...
  15. B

    What Is the Probability of Zero Cracks in 5 Miles of Highway?

    The number of cracks in a section of interstate highway that are significant enough to require repair is assumed to follow a Poisson distribution with a mean of two cracks per mile. What is the probability that there are no cracks that require repair in 5 miles of highway? any help guys? :)
  16. P

    MHB Poisson Distribution: Prob of <=3 Wrong Connections in Building

    A building has 2 independent automatc telephone exchanges A and B. The number X of wrong connections for A in anyone day is a poisson variable with parameter 0.5 and the number Y of wrong connections for B in any one day is a poisson variable with parameter 1. Calculate in any particular...
  17. L

    How Einstein field equation becomes the Poisson equation?

    I want to show that ∇2ϕ=ρ/2, which governs gravity in Newtonian physics? I found solution of this question in [General Relativity for Mathematicians, R.K.Sachs and H.Wu, 1997, page 112&271]. Solution refer to optional exercise as follows: Let R^ be the (0, 4) –tensor field physically...
  18. 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...
  19. 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...
  20. K

    Calculating Poisson Uncertainties with Changing Observation Durations

    I am trying to plot the flux from an Astrophysical source as a function of time. Due to the nature of the source, I am only receiving a handful of photons in each time bin. So imagine I had 10 observing periods of 10 days each, in which my telescope received the following number of photons...
  21. S

    Poisson Distribution - why are these different?

    Homework Statement X(t) is a Poisson process with \lambda=0.2 events per second. What is the probability of zero events in 45 seconds? 2. The attempt at a solution \frac{45}{0.2}=225 (0.2 second intervals) so P[X=0] in 225 consecutive intervals is: \left(e^{-0.2}\right)^{225} = 2.86...
  22. M

    How to find equations for confidence limits in Poisson distribution?

    Homework Statement What kind of equations you'll get when trying to find confidence limits 100(1-a) % for λ in Poisson distribution? Homework Equations Poisson distribution P(X=x) = e-λ λx / x! (x=0,1,2 ...) The Attempt at a Solution I made an equation as follows: Ʃ (k = from k0 to n) e-λ...
  23. S

    Charge Density Function to Solve Poisson Eq.

    Homework Statement This is not really a homework just studying but I'm kinda stuck. So I am trying to find out how to formally write down the Charge Density for any distribution. Although I will not get into Green's Function or how to find V, I got that fine. My example will be a Rod of...
  24. L

    Relation between Gamma and Poisson

    I'm having trouble doing a classic proof (integration par part and induction on r) for this : Pr(X>t)=Pr(Y ≤r−1), where X follows a gamma Γ(α = r, β = 1/λ) and Y a Poisson P (λt). Start with r = 1 (exponential distribution). I don't really understand what induction on r really means...
  25. I

    Not quite Poisson - how do i derive a transform pdf/cdf?

    Hello, I'm looking at some sporting data (similar to goals in a match) and trying to figure out what distribution applies to their count per match. Typically, Poisson is used in the industry to model the distribution. When I look at the historical events, poisson isn't too bad, but tends...
  26. T

    Probability - Poisson Random Variable?

    1. Homework Statement During a typical Pennsylvania winter, I80 averages 1.6 potholes per 10 miles. A certain county is responsible for repairing potholes in a 30 mile stretch of the interstate. Let X denote the number of potholes the county will have to repair at the end of next winter. 1...
  27. P

    MATLAB Poisson Equation with Mixed Boundary Condition - MatlaB

    Hi guys, I'm solving a Poisson Equation with Mixed Boundary condition. But I have trouBle with that mixed BC in MATLAB. Anyone can help to fix? Thanks a lot! dT^2/dx^2+dT/dy^2=-Q(x,y)/k Rectangular domain (HxL), BC: Top: T(x,H)=Th, Left: dT/dx=0, Bottom: dT/dy=q, Right: dT/dx+B(T-Tinf)=0...
  28. M

    Poisson Equation for a Scalar Field

    We all know that for the gravitational field we can write the Poisson Equation: \nabla^2\phi=-4\pi G\rho But I was wondering if, mathematically, we can write the same equation for a scalar field which scale as r^{-2}. Here is the thing. When you deal with gravity, the Poisson equation is...
  29. X

    Calculating Poisson Distribution for Telephone Calls in College Switchboard

    Homework Statement Telephone calls enter a college switchboard according to a Poisson process on the average of three calls every 4 minutes (i.e., at a rate of λ=0.75 per minute). Let W denote the waiting time in minutes until the second call. Compute P(W>1.5 minutes). Homework Equations...
  30. Z

    Is It Binomial or Poisson Distribution for Element Damages in Large Systems?

    I might need you guys to help me see how this proces, will be distributed: Suppose we have a large amount of elements N(≈1012). I'm simulating a system where I for each iteration damage a random element. If an element gets damaged its damagecounter goes up 1. So say I pick element number...
  31. Y

    Bio-statistics, Poisson distribution

    I'm studying Bio-statistics and I came across this problem from the textbook.It's actually answered on the back of the book, but I couldn't really get the same numbers. i Desease-free infants at the end of month i 0 2500 1 2425 2 2375 3 2300 4 2180 5 2000 6 1875 7 1700 8 1500 9...
  32. A

    How to Get Covariance of Bivariate Poisson Distribution

    Dear all, I have a problem in solving covariance of Bivariate Poisson Distribution Let X_i \sim POI (\theta_i) , i = 1,2,3 Consider X = X_1 + X_3 Y = X_2 + X_3 Then the joint probability function given : P(X = x, Y = y) = e^{\theta_1+\theta_2+\theta_3} \frac {\theta_1^x}{x!} \frac...
  33. D

    Poisson approximation distribution

    Homework Statement In a manufacturing process for electrical components, the probability of a finished component being defective is 0.012, independently of all others. Finished components are packed in boxes of 100. A box is acceptable if it contains not more than 1 defective component...
  34. S

    Poisson integral formula to solve other integrals

    Homework Statement Use 1) \frac{1}{2\pi}\int\limits_{-\pi}^{\pi} \frac{r_0^2 - r^2}{r_0^2 - 2rr_0cos(\theta-t) + r^2} dt = 1 to compute the integral: 2) \int\limits_{-\pi}^{\pi} \left[1 - acos(x) \right]^{-1} dx for 0<a<1 [/itex]. The Attempt at a Solution I looked on Wolfram...
  35. B

    Poisson Distribution of Accidents

    Homework Statement In New York in the last 3 years there were 55 driving accidents. Assume all days are alike. What is the approximate probability that "in the next 3 years there will be at least 2 days with more than one accident". Homework Equations Poisson approximation The...
  36. A

    Sum of two independent Poisson random variables

    Hello! I am trying to understand an example from my book that deals with two independent Poisson random variables X1 and X2 with parameters λ1 and λ2. The problem is to find the probability distribution of Y = X1 + X2. I am aware this can be done with the moment-generating function technique...
  37. N

    Why does the Poisson distribution apply here?

    Hello, I'm reading a text about statistics, but I don't understand why Poisson applies. (Note, this is not an assignment or anything like that.) Why would X be Poisson distributed with that parameter theta? The only Poisson that I could find reasonable is modelling X as Poisson...
  38. T

    Calculate Posterior p.f. from Poisson Distribution

    Homework Statement Suppose that the number of defects on a roll of magnetic recording tape has a Poisson distribution for which the mean λ is either 1.0 or 1.5, and the prior of λ is the following: L(1.0)=0.4 and L(1.5)=0.6 If the roll of tape selected at random is found to have 3...
  39. N

    Are Both Sensible Interpretations of Poisson Behavior?

    Are both sensible (equivalent? contradictory?) interpretations of "Poisson" behavior? I've come across two quite distinct notions (or so it seems to me, anyway) of Poisson behavior and I'm not sure if they're equally sensible or perhaps even equivalent. I'll apply both "views" to the same case...
  40. M

    Poisson MLE and Limiting Distribution

    Homework Statement Let yi denote the number of times individual i buys tobacco in a given month. Suppose a random sample of N individuals is available, for which we observe values 0,1,2,... for yi. Let xi be an observed characterisitc of these individuals (for example, gender). If we...
  41. maverick_starstrider

    Poisson Bracket to Commutator, What Does it REALLY Mean?

    Let me just head off the first waves of posts this thread will likely get. I am very fluent in quantum mechanics. I am completely aware of the behaviour of a commutator structure: simultaneous eigenbasis, etc. I understand how commutators model the structure that quantum mechanics has. My...
  42. A

    Poisson Kernel: Examining Half Plane Limit Case

    Homework Statement Can you look at Poisson's formula for a half plane as a limit case of Poisson's formula for a disk? http://en.wikipedia.org/wiki/Poisson_kernel I can find lots of information about the Poisson kernel for a disk, but not for the half plane. I do know on can mat the unit...
  43. J

    Poisson Distribution for Drill Stock Management

    Homework Statement A hardwhere store sells on average 8 drills per week. The store receives ONE delivery of drills at the same time each week. Find the no. of drills that need to be in stock after a delivery for there to be at most a 5% chance of the store NOT having sufficent drills to meet...
  44. 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...
  45. B

    Standard Benchmark Problem for Computational Solution of Poisson Equation

    Hi, I am working on FEM methods as a part of my senior year project and I have written a poisson solver for the same purpose. The solver works pretty well on the simple problems that I have designed as of now and seems to give correct answer (i.e. the data matches the theoretical prediction) 1...
  46. Z

    FEM software? Poisson equation: Boundary conditions for a charged boundary

    Hi all, I want to calculate the electrostatic potential for an two-dimensional area with given Dirichlet boundary conditions (say, a square) with a charged ring in it (like a wedding ring, but inifinitely thin) with a given line charge density. I figured out that the problem should be...
  47. 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...
  48. 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...
  49. P

    Poisson distribution and random processes

    Hello! I am writing because I recently became interested in probability distributions, and I have to you a few questions. Poisson distribution is given as a probability: f(k;\lambda)=\frac{\lambda^{k}e^{-\lambda}}{k!} But what is lambda? Suppose that we consider as an unrelated incident...
  50. C

    Drude Theory of Metals Poisson Distribution Problem

    Homework Statement This is the first problem from Ashcroft's Solid-State Physics which I recently picked up due to having far too much free time. The first two parts of the problem relate to the probability that an electron picked at random will have had no collision during the preceding t...
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