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. maverick280857

    Basic question about the generalized Poisson Equation

    Hi, Suppose we look at two dimensional Poisson's equation in a medium with spatially varying (but real) dielectric constant: \nabla(\epsilon_r\nabla \varphi) = -\frac{\rho(x,y)}{\epsilon_0} Consider the problem of solving this using the Finite Difference method on a rectangular grid...
  2. A

    Transforming a % variation of the mean from Poisson to σ

    Hi! I do have this problem - Consider that for a set of values, I do have a Poisson distribution with mean value <m> - Now, I need to gather another set of dataset, which I should vary the mean value by 5% - My question is, how can I translate each one of these new values to sigma deviations...
  3. S

    Poisson brackets for simple harmonic oscillator

    Homework Statement Considering the Hamiltonian for a harmonic oscillator: H=\frac{p^2}{2m}+\frac{mw^2}{2}q^2 We have seen that the equations of motion are significantly simplified using the canonical transformation defined by F_1(q,Q)=\frac{m}{2}wq^2cot(Q) Show explicitly that between both...
  4. H

    MHB Uniformity of Poisson arrivals in random interval

    Given that an Poisson arrival has occurred in an interval [0,t], where t is geometric with mean (alpha). Is it true that the arrival instant is uniform in [0,t]?
  5. G

    Calculating Variance of a Sum of Independent Random Variables

    Homework Statement During a two hour window, people are given the option of calling number X, donating $9.90, or number Y, donating $0.50. X is Poisson distributed with 1500 calls/minute. Y is Poisson with 3750 calls/minute. What is the probability that more than $2,000,000 is raised...
  6. K

    Help Integrating Poisson Errors on Histograms

    So I have a histogram with bins that contain the number of events expected at a specific energy (which I generated with a Monte Carlo).. I need to add (integrate) all the bins in a section of this histogram and find the error of this value. However, the number of events are very small approx...
  7. P

    Statistics - Poisson distribution question.

    Hi, Homework Statement I am somewhat perplexed by the proposed solution to the following Statistics problem and was hoping one of you might be willing to help me settle this: An operator receives phone calls between 8AM and 4PM at an average rate of 20 calls/hour. No call was received during...
  8. S

    Defining Poisson Brackets: Analytic Functions in Multiple Variables

    l know you can define poisson brackets between two analytic function in several variables f(q1,q2,q3,..,p1,p2,p3,..) and g (q1,q2,q3,..,p1,p2,p3,..) only by foundamental poisson brackets and their proprieties.how is it possible?
  9. stripes

    Proving the Poisson Summation Formula: A Formal Approach

    Homework Statement Prove the Poisson summation formula. Homework Equations The Attempt at a Solution Correction to image below: the very last line of the theorem (italicized) should say f hat is the Fourier transform, not f(n). Does this proof make sense and is it...
  10. 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...
  11. binbagsss

    Radioactive Decay - Gaussian or Poisson

    Radioactive Decay Probability? Say you are counting the number of decays and the time of observation is varied. I know that as time increases, the Gaussian Distribution becomes a closer fit to the observed probability than when the time interval takes smaller values because the mean count...
  12. D

    Constant source Poisson eq in 2D, Dirichlet BC, average value?

    Hello, for the Poisson problem Δu = -1 on a 2D circular disk with u = 0 on the boundary, we have average(u) = \frac{1}{8\pi}Area(disk), which is easy to see, as the solution is quadratic in the polar coordinate r. Does this (or a similar) relation hold for non-circular 2D domains? This...
  13. dkotschessaa

    Probability - Poisson Probability

    Homework Statement Show that the Poisson probabilities p_{0}p_{1},... can be estimated recursively by p_{0} = e^{-\lambda} and p_{k}=(\lambda/k)*p_{k-1} k=1,2,... Homework Equations I know the Poisson distribution f(x, \lambda) = e^{-\lambda}\lambda^{x}/x! But I...
  14. 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...
  15. twoski

    Probability and Poisson Random Variable

    Homework Statement A trial consists of throwing two dice. The result is counted as successful if the sum of the outcomes is 12. What is the probability that the number of successes in 36 such trials is greater than one? What is this probability if we approximate its value using the Poisson...
  16. J

    Poisson vs Binomial distribution.

    Hello PF This might be a fairly simple question to most of you, but I was given this problem (don't worry, I already solved it just wondering about something) Suppose the probability of suffering a side effect of a certain flu vaccine is 0.005. If 1000 persons are inoculate, find the...
  17. N

    Neuroscience: poisson and gauss in neuron firing rate model

    Hello! I was reading a journal article on modeling the interaction between different neural networks and I am confused about the follwoing method (cited below). It is describing the spike rate output of a neuron based on oscillating firing rates of excitatory (E) and inhibitory (I) inputs...
  18. J

    Binomial vs Poisson Distributions

    Homework Statement I was given two problems and required to calculate some statistics/parameters for them. They are: 1) The Vancouver Island Marmot is one of Canada’s most endangered species; there are currently only 63 animals left on the Island. To maintain the population, geneticists...
  19. V

    Poisson Distribution Statistics

    Homework Statement If the number of complaints a dry cleaning establishment receives per day is random variable having the Poisson distribution with λ = 3.3, what are the probabilities that it will receive: (a) Five complaints altogether on any two given days. (b) at least 12 complaints...
  20. W

    Statistics: Poisson Distribution

    1. A Poisson random variable is such that it assumes the values 0 and 1 with equal probability. Find the value of the Poisson parameter, ρ ,for this variable. 2. Poisson equation: f(x) = e-λs(λs)/x! 3. I assumed the probability would be 0.5 because it can be either 0 or 1. 0.5 = e-λs(λs)/x! But...
  21. B

    Poisson Bracket for 1 space dimension field

    Hi, Suppose you have a collection of fields \phi^i (t,x) depending on time and on 1 space variable, for i=1,...,N. Its dynamics is defined by the Lagrangian L=\frac{1}{2} g_{ij}(\phi) (\dot{\phi}^i \dot{\phi}^j - \phi ' ^i \phi ' ^j ) + b_{ij}(\phi) \dot{\phi}^i \phi ' ^j where...
  22. L

    Are Poisson Brackets Consistent in Electromagnetic Field Theories?

    Since I couldn't find any reference on the subject of Poisson bracket formalism of classical field theory, I'm posting a few question here: A) What are the Poisson brackets of the source-less EM field? B) Does the law that the Poisson brackets between a dynamical variable and its conjugate...
  23. B

    Poisson brackets of angular momentum components

    I want to find [M_i, M_j] Poisson brackets. $$[M_i, M_j]=\sum_l (\frac{\partial M_i}{\partial q_l}\frac{\partial M_j}{\partial p_l}-\frac{\partial M_i}{\partial p_l}\frac{\partial M_j}{\partial q_l})$$ I know that: $$M_i=\epsilon _{ijk} q_j p_k$$ $$M_j=\epsilon _{jnm} q_n p_m$$ and so...
  24. P

    Expressing sol. of Poisson eqn. in terms of vol. and sur. integrals.

    Hi, Referring to Jackson's Electrodynamics 3ed, page 197, line 5. He assumes that the magnetization can be divided into volume part and surface part, thus generating eqn 5.100. This is fine. In a straightforward way, I wanted to do the same but for electrostatics, eqn 4.32:∅= (1/4πε) ∫dv...
  25. C

    2-D Poisson Equation Boundary Value Prob

    Homework Statement Solve the equation: ∂2F/∂x2 + ∂2F/∂y2 = f(x,y) Boundary Conditions: F=Fo for x=0 F=0 for x=a ∂F/∂y=0 for y=0 and y=b Homework Equations How can I find Eigengunctions of F(x,y) for expansion along Y in terms of X? The Attempt at a Solution I can't imagine...
  26. C

    What is the Poisson Approximation for Small Probability Trials?

    Homework Statement Consider ##n## independent trials, each of which results in one of the outcomes ##1,...k## with respective probabilities ##p_1,...p_k, \sum_{i=1}^{k} p_i = 1##. (I interpret this summation as just saying mathematically that definitely one of the outcomes has to occur on each...
  27. M

    Exploring the Impact of Poisson Kernel on Math Functions

    Why Poisson kernel is significant in mathematics? Poisson kernel is ##P_r(\theta)=\frac{1-r^2}{1-2rcos\theta+r^2}##. http://www.math.umn.edu/~olver/pd_/gf.pdf page 218, picture 6.15. If we have some function for example ##e^x,sinx,cosx## what we get if we multiply that function with Poisson...
  28. A

    Gamma Poisson Mixture with finite Gamma

    Dear all I am working with a Gamma-Poisson mixture distribution where (and this is not usual) the support of the Gamma distribution (in fact, I am able to restrict to the neg exponential instead of the Gamma...) is finite, e.g. [0,1]. I would like to derive the mean and a Likelihood...
  29. M

    MLE of Poisson Dist: Find \lambda^2+1

    Homework Statement Let X_1,...,X_n be a random sample from a poisson distribution with mean \lambda Find the MLE of \lambda^2 + 1 Homework Equations The Attempt at a Solution I found \hat{\lambda}=\bar{x} Can I just square it and add 1 and solve for lambda hat? If not I have no idea...
  30. O

    Independent Poisson Processes Word Problem

    Hello, hopefully this is the right place. This is a homework question, so it should definitely be in this forum, but I wasn't sure which sub-forum to put this rather elementary stats question. Homework Statement In my introductory mathematical statistics class, we've been given the...
  31. S

    Poisson distribution questions

    Homework Statement Suppose x has a Poisson \lambda distribution Find the probability generating function and range it is well defined. Then evaluate E[x(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-7)(X-8)(x-9)(x-10)(x-11)] Homework Equations f_x (x) = exp(-lamda) (lamda)^x/x! for...
  32. A

    Poisson and Gamma Distributions

    Let Y|X be a Poisson(X), and X be Gamma(\alpha, \beta). Find E(X|Y)... Since Y|X is Poisson(X), we have f(Y|X)= \frac{m^x e^{-m}}{x!}... Since X is Gamma(\alpha, \beta), we have f(x)= \frac{x^{\alpha-1} e^{-x/B}}{\Gamma(\alpha) \beta^{\alpha}}... Since f(Y|X) = \frac{f(x,y)}{f(x)} ====>...
  33. A

    Understanding Poisson Brackets in Symplectic Notation

    Okay there is a particular equation in my book, which I just can't seem to understand intuitively. I've been staring at it for an hour now without progress, so I hope some of you can explain it. Basically it's the one on the attached picture. Let me introduce the notation so you can help me...
  34. D

    Statistics, Poisson processes.

    Homework Statement Homework Equations The Attempt at a Solution Here's what I've tried so far, not really sure how to go on with these problems. Been reading the textbook up and down on Poisson processes! Any hints or help? Especially 1.B, 1.C and 2.B, 2.C
  35. C

    Help with a non-homogeneous poisson distribution please

    Hello, I want to be able to model something with a poisson process with an intensity function that changes with both time and space. Let's say for example that the time interval I'm considering is 100 hours long and I believe that the intensity function increases at a constant rate so that...
  36. D

    MHB Poisson kernel significance

    What is the significance of the Poisson kernel (besides solving the Dirichlet problem)? What is the Poisson's role in solving the Dirichlet problem? I know it is the solution but what is meant by its role?
  37. D

    MHB What is the Limit of the Poisson Kernel Prove for $r\to 1$?

    Prove: $$ \lim_{r\to 1}P(r,\theta) = \begin{cases} \infty, & \theta = 0\\ 0, & \text{otherwise} \end{cases} $$ For the first piece, take the summation $$ P(1,0) = \frac{1}{\pi}\left(\frac{1}{2} + \sum_{n = 1}^{\infty} 1^n\right). $$ Then $\sum\limits_{n = 1}^{\infty} 1^n = \infty$. Therefore, we...
  38. D

    MHB Prove Evenness of Poisson Kernel for Fixed $r$

    For a fixed $r$ with $0\leq r < 1$, prove that $P(r,\theta)$ is an even function.Take $-r$. Then \begin{alignat*}{3} P(-r,\theta) & = & \frac{1}{2\pi}\frac{1 - (-r)^2}{1 - 2(-r)\cos\theta + (-r)^2}\\ & = & \frac{1}{2\pi}\frac{1 - r^2}{1 + 2r\cos\theta + r^2} \end{alignat*} I have $1 +...
  39. jfy4

    Poincare Algebra from Poisson Bracket with KG Action

    Homework Statement Consider the Klein-Gordan action. Show that the Noether charges of the Poincare Group generate the Poincare Algebra in the Poisson brackets. There will be 10 generators.Homework Equations \{ A,B \}=\frac{\delta A}{\delta \phi}\frac{\delta B}{\delta \pi}-\frac{\delta...
  40. C

    Proving the Poisson summation formula (like a physicist)

    Hi! I'n my quantum mechanics homework I've been asked to proved the Poisson summation formula. The mathematicians seem to use abstract and confusing notation when proving this kind of thing so I'm hoping for some help from physicists in standard notation ;) I'm starting with a function f(x) =...
  41. fluidistic

    Moments of the Poisson distribution

    I cannot seem to get the first moment of Poisson's distribution with parameter a: P(n_1)=\frac{a^{n_1}e^{-a}}{n_1!} when using the characteristic function \phi _X (k)=\exp [a(e^{ik}-1)]. The definition of the first moment involving the characteristic function is <n_1>=\frac{i}{n} \frac{d \phi...
  42. G

    Question about how to merge poisson distribution

    In general, if A~Po(a) and B~Po(b) are independent random variables, then C = (A+B)~Po(a+b). Can someone please explain the intuition/simple proof of this and a word problem or example would really help to reinforce this concept. Thanks.
  43. R

    Poisson brackets and angular momentum

    Homework Statement Let f(q, p), g(q, p) and h(q, p) be three functions in phase space. Let Lk = εlmkqlpm be the kth component of the angular momentum. (i) Define the Poisson bracket [f, g]. (ii) Show [fg, h] = f[g, h] + [f, h]g. (iii) Find [qj , Lk], expressing your answer in terms of...
  44. J

    Why does Binomial dist. converge in distribution to Poisson dist. ?

    Hey guys, In class, I was shown that the Binomial prob density function converges to the Poisson prob density function. But why does this show that the Binomial distribution converges in distribution to the Poisson dist. ? Convergence in distribution requires that the cumulative density...
  45. V

    Poisson Bracket - Constrained system

    Hi friends I am trying to drive constraints of a Lagrangian density by Dirac Hamiltonian method. But I encountered a problem with calculating one type of Poisson Bracket: {\varphi,\partial_x\pi}=? where \pi is conjugate momentum of \varphi. I do not know for this type Poisson Bracket I can...
  46. A

    Poisson and continuity equation for collapsing polytropes

    Hello everybody! I am using in my studies this beautiful book by Kippenhahn & Weigert, "Stellar Structure and Evolution", but I have some problems about collapsing polytropes (chapter 19.11)... After defining dimensionless lenght-scale z by: r=a(t)z and a velocity potential \psi...
  47. G

    Mean of Poisson Dist, given mode

    Suppose a person takes data (say counts per minute of cars going past his window), for a long time. Then he loses his data, but knows that he counted 5 cars more often than any other number. What is the likely range for the average count rate? I tried to solve this by saying the mode is 5, so...
  48. S

    Poisson distribution help

    1. The number of times that a person contracts a cold in a year is a Poisson random variable with parameter lambda=5. Suppose a wonder drug reduces the Poisson parameter to lambda=3 for 75% of the population but does not affect the rest of the population. If an individual tries the drug for a...
  49. 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...
  50. T

    Poisson Distribution Help

    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...
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