What is Poisson: Definition and 507 Discussions

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. S

    Poisson distribution problem

    Homework Statement We assume that the number of structural flaws on a long wire have obey Poisson distribution law. On average we find 1 flaw every 5 meters. a) What is the probability that a 20 m long section will have maximum 2 flaws? b) We slice the wire into 1 m long sections. What is the...
  2. C

    Probability of Poisson event happening twice, consecutively

    Homework Statement The number of telephone calls, T, received each minute can be modeled by a Poisson distribution with a mean of 3.5. Find the probability that at least three telephone calls are received in each of two successive one-minute intervals. Homework Equations P =...
  3. C

    MHB Probability of event modelled by poisson happening twice, consecutively

    I'm not great at statistics, so I don't know where to start with this problem. It is stated as follows: The number of telephone calls, T, received each minute can be modeled by a Poisson distribution with a mean of 3.5. Find the probability that at least three telephone calls are received in...
  4. T

    Solving Poisson Distribution Problems: Questions on Calls/Minute

    Homework Statement A telephone operator receives four phone calls in three minutes on the average. Let a Poisson random number X denote the number of phone calls per minute to this operator. (a) Find the probability that this operator receives two phone calls in a minute. (b) Find the...
  5. sunrah

    Binomial -> Poisson distribution question

    Homework Statement A teacher has an infinite flow of papers to mark. They appear in his office at random times, at an average rate of 10 a day. On average 10% of the manuscripts are free from errors. What is the probability that the teacher will see exactly one error-free manuscript (a) after...
  6. D

    When/how to reject Poisson distribution hypothesis?

    Homework Statement I have run into a situation that my gut tells me is impossible (alright extremely unlikely) when assuming a Poisson distribution. I want to make this gut feeling more formal by testing it against a Poisson distribution. Sadly I'm not a schooled statistician. Generalised...
  7. C

    Poisson Probability Distribution Problem

    Homework Statement An article suggests that a Poisson process can be used to represent the occurrence of structural loads over time. Suppose the mean time between occurrences of loads is 0.4 year. a). How many loads can be expected to occur during a 4-year period? b). What is the probability...
  8. S

    Poisson equation in R with a source at the origin

    Homework Statement Solve the poisson eq. on R with a source in x=0. The Attempt at a Solution I haven't done this kind of thing in years, so I'm a bit rusty, but I think that this is requested: \Delta \phi = - \rho \delta(x) (Edit: no wait, I need an integral here). It doesn't seem to be...
  9. hideelo

    Q about Poisson eqn w/ Neumann boundary conditions as in Jackson

    I am reading Jackson Electrodynamics (section 1.10 in 3rd edition) and he is discussing the Poisson eqn $$\nabla^2 \Phi = -\rho / \epsilon_0$$ defined on some finite volume V, the solution using Greens theorem is $$\Phi (x) = \frac{1}{4 \pi \epsilon_0} \int_V G(x,x') \rho(x')d^3x' +\frac{1}{4...
  10. S

    Exercise on Poisson distribution

    Homework Statement An experimenter measures the counting rate from a radioactive source as 10,150 counts in 100 minutes. Without changing any of the conditions, the experimenter counts for one minute. There is a probability of about 15 percent that the number of counts recorded will be fewer...
  11. R

    Fundamental Poisson Bracket - Canonical Transformation

    When proofing the poisson brackets algebraically, what is the tool of choice. Can one use the muti dimensionale chain rule or how is it usally done?
  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. Coffee_

    Poisson equation with a dirac delta source.

    Consider: ##\nabla^{2} V(\vec{r})= \delta(\vec{r})## By taking the Fourier transform, the differential equation dissapears. Then by transforming that expression back I find something like ##V(r) \sim \frac{1}{r}##. I seem to have lost the homogeneous solutions in this process. Where does this...
  14. O

    Test set-up for full dark matter simulation

    Hey, I'm trying to look for a single test set-up for a dark matter-only simulation I and my friend are building. It's currently based on a particle-in-cell approach and we are calculating the Poisson equation and particle trajectories in a co-moving frame (so expansion of the Universe is taken...
  15. D

    What is the physical significance of Poisson brackets?

    I know the definition of the Poisson bracket and how to derive elementary results from it, but I'm struggling to understand intuitively what they are describing physically? For example, the Poisson bracket between position q_{i} and momentum coordinates p_{j} is given by \lbrace...
  16. mishima

    52 card Poisson Distribution experiment?

    Hi, I was trying to think of a way to generate a Poisson distribution using a single deck of 52. Say I am looking at the position of the Ace of spades in the deck after a number of shuffle rounds (1 shuffle round is 7 riffle type shuffles). Success is that an Ace of spades is on top of the...
  17. H

    Looking for a modified Poisson distribution

    I'm looking to model a system in which events are nearly perfectly randomly distributed but with a slight tendency for events to avoid each other. As you know, if the system were perfectly random, I could use a Poisson distribution. The probability distribution for the number of events would...
  18. M

    I am having difficulty with a Gamma Distribution problem

    Homework Statement Automobile accidents occur in the United States over a 72 hour holiday period like events in a Poisson process with parameter lambda=10/hr. V is the time until the 10th accident a) what is expected value of V or E[V] and standard deviation? b) What is the probability that V...
  19. H

    How did the Poisson brackets get derived, and from what.

    do they have a physical meaning and did they fall out of another theory. I have only ever seen them stated as a fact, I am assuming they are a result of something ie a consequence of another more fundamental theory. when are they used in a practical problem solving sense to solve real world...
  20. T

    General Solution of a Poisson Equation of a magnetic array

    Hi, I'll give some background, say you've got a planar structure of thickness 'd', lying on the z plane. Also say the upper and lower surfaces are y = 0 and y = -d, respectively. The structure has scalar potentials inside it as so: As you can see the vector fields cancel out on one side, As it...
  21. F

    Poisson equation/zero padding and duplicating Green function

    Hello, I need to solve the Poisson equation in gravitational case (for galaxy dynamics) with Green's function by applying Fast Fourier Transform. I don't understand the method used for an isolated system from (Hockney & Eastwood 1981); it says : I have 2 questions: * Why we duplicate the...
  22. T

    General Solution of a Poisson Equation (maybe difficult)

    Hi, This is overwhelmingly more of a maths problem than a physics problem, because it's all theoretical. I'll give some background to modle it incase the math's isn't enough. Say you've got a planar structure of thickness 'd', lying on the z plane. Also say the upper and lower surfaces are y = 0...
  23. K

    Poisson PDE in polar coordinates with FDM

    I want to solve a Laplace PDE in a polar coordinate system with finite difference method. and the boundary conditions: Here that I found in the internet: and the analytical result is: The question is how its works? Can I give an example or itd?Thanks
  24. A

    MHB Wrecking my brains on this Poisson Distribution question.

    Hi. I normally can solve poisson distribution questions with ease. But this one question had me thinking for hours on end with no solution. It would be great if someone can help me. QN: The number of incoming calls per minute, X, to a telephone exchange has a Poisson distribution with mean 2...
  25. rayne1

    MHB Poisson distribution problem

    Problem: McBurger’s drive-thru has only one service window and serves an average of 2 customers every 5 minutes. 70% of customers order drinks from the drive-thru. The manager monitors the employee at the drive-thru for the next 3 hours. He will give the employee a raise if exactly 20 customers...
  26. 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...
  27. C

    MHB Examples of uses for the Poisson Eqn in 1d

    Hi all, I have almost finished my dissertation on using the finite element method to solve the 1D version of the Poisson equation. For the last section I would like to run through a couple of examples but am struggling to find some. Obviously I can make up any equations that satisfy the...
  28. Coffee_

    Canonical transformations, poisson brackets

    Three questions1) Let's say that N ##q_i## and ##p_i## are transformed into ##Q_k## and ##P_k##, so that: ##q_i = q_i(Q_1,Q_2,. ... , P_1,P_2, ... ) ## and ##p_i=p_i((Q_1,Q_2,. ... , P_1,P_2, ... )## We have proved that these transformations are canonical only and only if ##\forall i##...
  29. C

    Probability Conditional Expectation

    Suppose X and Y are independent Poisson random variables with respective parameters λ and 2λ. Find E[Y − X|X + Y = 10]3: I had my Applied Probability Midterm today and this question was on it. The class is only 14 people and no one I talked to did it correctly. The prof sent out an e-mail saying...
  30. JonnyMaddox

    Infinitesimal transformations and Poisson brackets

    Hello, I want to understand how bracket operations in general are related to symmetry and infinitesimal transformations (in hindsight of quantumfieldtheory), so I calculated an example with a particle that is moving on a circle with a generic potential. (I used simple polar coordinates in two...
  31. K

    Poisson equation for the field of an electron

    Homework Statement In classical electrodynamics, the scalar field \phi(r) produced by an electron located at the origin is given by the Poisson equation \nabla^2\phi(r) = -4\pi e\delta(r) Show that the radial dependence of the field is given by \phi(r) = \frac er Homework Equations I'm not...
  32. Jameson

    MHB Estimating variance of Poisson random variable

    I am trying to use a generated random sample in R to estimate the mean and variance for a Poisson random variable. The first one is a Poisson random variable with mean 5. To estimate the above I generate a random sample in R with the following code: P5 <- rpois(100,5) Given the above I want to...
  33. 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...
  34. 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...
  35. N

    How can ISO 11929 norm help combine Poisson errors in low counting statistics?

    I have low counting stats and need to subtract background, account for efficiency, and divide by volume. How do I combine the asymmetrical (Poisson) errors?
  36. binbagsss

    Stats - mle poisson distribution -- quick question

    This is probably a stupid question , but, It's easy enough to show that the mle of a poission distribution is ## \bar{x}##: ## \hat{ \lambda}= \bar{x} ## But,I'm then looking at the generalized ratio test section of my book, multinomial, it esitmates ## \lambda ## for some data by ## \sum...
  37. C

    Separation of Variables Spherical Coordinates

    Homework Statement So I'm doing a question from one of my past exams as attached, there are no copy right issues with this document that I know of and have asked my lecturer who wrote the exam and he said I am welcome to upload it. The question is 1)b)iv), my attempt is attached. I end up with...
  38. M

    Poisson, Binomial Distributions

    Homework Statement The number of claims that an insurance company receives per week is a random variable with the Poisson distribution with parameter λ. The probability that a claim will be accepted as genuine is p, and is independent of other claims. a) What is the probability that no claim...
  39. W

    Poisson's ratio for a rigid rod

    I have a conceptual misunderstanding it seems. Poisson's ratio is the ratio of elastic strain deformation of the transverse and longitudinal components. That being said, if I were to induce thermal stress (heating up) to a rod by keeping its ends (longitudinal component) rigid, would there be a...
  40. M

    Help with Poisson Brackets

    Homework Statement Consider the motion of a particle with charge e in a homogenous magnetic field B_i. The Hamiltonian for this problem is $$H = \frac{1}{2m} \sum_{i=1}^{i=3} \left[ p_i - \frac{e}{2}\epsilon _{ijk}B_j x_k\right]^2.$$ By calculating the Poisson brackets, show that the...
  41. O

    Poisson and the integral of motion

    I am stuck on a proof. I need to show that if a Hamiltonian only depends on q1 and p1 though a function f(q_1,p_1), that is; H(f(q_1, p_1), q_2, p_2, q_3, p_3, ... q_n, p_n) then f(q_1, p_1) is an integral of motion. My attempt at a solution is as rather simplistic but I'm stuck making the...
  42. B

    Finding the conditional distribution

    Hey guys, I'm trying to find a conditional distribution based on the following information: ##Y|u Poisson(u \lambda)##, where ##u~Gamma( \phi)## and ##Y~NegBinomial(\frac{\lambda \phi}{1+ \lambda \phi}, \phi^{-1})## I want to find the conditional distribution ##u|Y## Here's what I've got so...
  43. A

    Poisson equation with three boundary conditions

    I have the following 2D Poisson equation (which can also be transformed to Laplace) defined on a triangular region (refer to plot): \begin{equation} \frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}=C\end{equation} with the following three boundary conditions...
  44. R

    2D cylindrical heat equation

    Hello friends, I am new for numerical methods and programming. i have been trying to devolop a program in 2D poisson heat equation in cylinder (r,angle) by finite difference method ∂2u/∂r2 + 1/r * ∂u/∂r + 1/r2 * ∂2u/∂θ2 = Q(u,θ)discritized equation :- ui+1,j − 2uij + ui−1,j/(∆r)2 + 1/ri *...
  45. K

    Poisson Summation in Heat Equation (Polar Coordinates)

    Homework Statement I'm currently trying to follow a derivation done by Shankar in his "Basic Training in Mathematics" textbook. The derivation is on pages 343-344 and it is based on the solution to the two dimensional heat equation in polar coordinates, and I'm not sure how he gets from one...
  46. A

    Have a software that solves Helmholtz equation, can I use it for Poisson?

    I only took one class of PDE and even though I do remember the relationship between Laplace and Poisson I really do not recall Helmholtz at all. Anyways, I am trying to figure out if my software (a software I found online, FISKPACK) that solves Helmholtz equation can be used to solve Poisson...
  47. T

    MHB Binomial probability or poisson?

    This the only question I'm having issues with. It may be a binomial distribution or poissm, not really sure. If an airplane has 224 seats and the no show rate of passengers with reservations is .09 how many reservations should the airline book such that the probability of not enough seats for...
  48. R

    Poisson 2D non linear by FDM

    Hi friends, i have developed an code for a non linear heat conduction in 2 dimensions with dirichlet boundary condition by finite difference method in Matlab. my code is running slow to give output. If anybody has any idea of solving this equation or have written any Code for this equation...
  49. D

    MHB Second moment of the Poisson random variable

    With a Poission random variable, we know that \(E[X] = var(X) = \lambda\). By definition of the variance, we can the second moment to be \[ var(x) = E[X^2] - E^2[X]\Rightarrow E[X^2] = var(X) + E^2[X] = \lambda(1 + \lambda). \] The characteristic equation for the Poisson distribution is...
  50. 0

    How do I calculate this Poisson bracket in QED?

    Homework Statement To calculate a certain Dirac bracket I need to calculate this Poisson bracket (Weinberg QTF 1 p.349 first eq.) $$[F,\Pi_i(\mathbf{z})]_P$$ where F is any functional of matter fields and their conjugates and pi is the conjugate to the vector potential. It should be zero...
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