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. T

    Poisson distribution ( approximation)

    Homework Statement The number of flaws in a plastic panel used in the interior of cars has a mean of 2.2 flaws per square meter of panel . What's the probability that there are less than 20 surface flaws in 10 square meter of panel ? Homework EquationsThe Attempt at a Solution This is a...
  2. 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...
  3. Danny Boy

    A Quantum measurement operators with Poisson distribution

    The following is a somewhat mathematical question, but I am interested in using the idea to define a set of quantum measurement operators defined as described in the answer to this post. Question: The Poisson Distribution ##Pr(M|\lambda)## is given by $$Pr(M|\lambda) =...
  4. P

    Poisson Random Variable probability problem

    Homework Statement X is a Poisson Random Variable with rate of 1 per hour, following the Poisson arrival process a. Find the probability of no arrivals during a 10 hour interval b. Find the probability of X > 10 arrivals in 2 hours c. Find the average interarrival time. d. For an interval of 2...
  5. King_Silver

    How Does the Poisson Distribution Calculate Tornado Probabilities in Georgia?

    Homework Statement The number of tornadoes per year, in Georgia, has a Poisson distribution with a mean of 2.4 tornadoes. Calculate the probability that in any given year, there will be: (i) At most 2 cases. (ii) At least one case. (iii) Calculate the probability that there will be...
  6. N

    I Poisson distribution regarding expected distance

    Hi The question is about diseased trees in an area (Poisson process), and states that λ = 15 diseased trees in a km square. I need to calculate expected distance from a point in the square to a diseased tree. Now I thought that this means that P(diseased tree = 0) ~ Po(15) = 3.059 x 10^-7 Or...
  7. U

    MHB Simple question regarding linear regression model poisson

    The question: Suppose $Y$ is discrete and only takes on non-negative integers and that the conditional distribution of $Y$ given $X=x$ is Poisson, that is, $$P(Y=y|X=x) = \frac{\exp(-x'\beta) (x'\beta)^y}{y!}$$ where $y = 0, 1, 2, \cdots$. First compute $E(Y|X=x)$ and $Var(Y|X=x)$, does this...
  8. L

    Poisson's equation in 1D with point source

    Homework Statement Solve ##\Delta\phi = -q\delta(x)## on ##\mathbb{R}##. Correct answer: ##\phi = -\frac{q}{2}|x| + Ax + B## Homework EquationsThe Attempt at a Solution In one dimension the equation becomes ##\frac{d^2 \phi}{d x^2} = -q\delta(x)##. We integrate from ##-\infty## to ##x## to...
  9. T

    Performing Metropolis-Hastings algorithm for a Poisson Distribution

    Homework Statement The number of busy lines in a trunk group (Erlang system) is given by a truncated Poisson distribution. I am asked to generate values from this distribution by applying the Metropolis-Hastings algorithm. Homework Equations The distribution is given in the attached picture...
  10. C

    A Cylindrical Poisson equation for semiconductors

    In a cylindrical symmetry domain ## \Phi(r,z,\alpha)=\Phi(r,z) ##. Does anyone can point me what can be found in literature to solve, even with an approximate approach, this equation? \nabla^2 \Phi(r,z)=-\frac{q}{\epsilon} \exp(-\frac{\Phi(r,z)-V}{V_t}) Where ## q, \epsilon, V ## and ## V_t ##...
  11. A

    Classical Please recommend two textbookss about the Poisson equation and Green's function

    Please recommend two textbooks about Poisson equation, Green's function and Green's theorem for a theoretical physics student. One is easy to read so that I can have an overall understanding of the topics, another is mathematically rigorous and has a deep and modern exploration of these topics...
  12. L

    I Is this a Poisson distribution problem?

    Hello, I have been thinking about this problem for a while, but I can't decide how it should be tackled statistically. I wonder if you can help, please. Suppose that prostheses for hip replacement are sold mainly by 2 manufacturers, A and B. Since they started being sold 20 years ago, 100 000...
  13. FallenApple

    A Interpreting Poisson Regression Estimates across groups

    Say for example I want to see the rate of injury for firefighter vs police vs soldier. ##InjuryCount_{i}## The number of injuries recorded for the ith person over time ##T_{i} ## Time the person was followed. Varies from person to person. ##I(f)_{i}## indicator for ith person of being a...
  14. S

    I Calculating Probability with Poisson Distribution: Radioactive Source Counts

    Hello! I came across this problem: A counter near a long-lived radioactive source measures an average of 100 counts per minute. What is the probability that more than 110 counts will be recorded in a given one-minute interval? I am not sure how to do it. Applying the Poisson distribution formula...
  15. J

    MHB How Do You Find the MLE of Lambda in a Sum of Two Poisson Distributions?

    First of all I will use L to denote lambda the parameter of the distribution. X~Poission(nL), n$\in\Bbb{N}$, Y~Poisson(mL),m$\in\Bbb{N}$ with m$\ne$n S= aX+bY a,b real constants. Given observations x and y find the maximum likelihood estimator of L. The problem is I don't know what the pmf...
  16. R

    Statistics probability questions

    Homework Statement Each week, Stéphane needs to prepare 4 exercises for the following week's homework assignment. The number of problems he creates in a week follows a Poisson distribution with mean 6.9. a. What is the probability that Stéphane manages to create enough exercises for the...
  17. A

    I Dimensional Analysis Poisson Equation

    Suppose I am given some charge density profile ρ(x). Poisson's equation in 1D reads d2φ/dx2 = ρ(x)/ε Is there a simple way to see what the order of magnitude of the electrostatic potential should be from a dimensional analysis?
  18. I

    I Experimental Shear Modulus and Poisson Ratio

    I'm designing a 'thing' for my masters project, which has to be 3D printed. I've designed this in Inventor, but it doesn't have PLA (Polylactic Acid), so I've been using Inventor's PMMA (Perspex) thus far. I want to get more accurate figures, so I've had a 1 cm3 block printed to get an exact...
  19. Mehmood_Yasir

    A Waiting time in a Queue using Poisson arrival

    Dear colleagues. I have a difficulty in calculating waiting time of the first packet arriving in a buffer using aggregation. Arrival follow Poisson process with rate lambda and the aggregation is done for a maximum 'm' packets or with the expiry of timer T_maximum. Can anyone help me how to...
  20. hikari1987

    A Exact solution to Poisson equation in 2D

    Hi all , Could you please help me solve Poisson equation in 2D for heat transfer with Dirichlet and Neumann conditions analytically? Thank you
  21. S

    Calculating Probability using the Poisson Distribution

    Homework Statement On average, 2 students per hour come into the class. What is the probability that the time between two consecutive arrivals is in the interval <10 minutes; 50 minutes>. Homework Equations p(k)=P(Y=k)=((lambda*t)k*(e-lambda*t)/k! The Attempt at a Solution I've tried using...
  22. J

    I Solving the 3D Poisson Equation Using Finite Difference/Volume

    Hi, I'm attempting to solve the 3D poisson equation ∇ ⋅ [ ε(r) ∇u ] = -ρ(r) Using a finite difference scheme. The scheme is simple to implement in 3D when ε(r) is constant, and I have found an algorithm that solves for a non-constant ε(r) in 2D. But I am having trouble finding an algorithm...
  23. mertcan

    Prob/Stats Exponential and Poisson distributions or processes

    hi everyone initially I really want to put into words that there is absolutely no source related to following probability in poisson process and distribution $$P(S^1_A<S^1_B<S^1_C)$$ or $$P(S^n_A<S^m_B<S^k_C)$$ where $$S^1_A = \text{first arrival of A event}, S^1_B= \text{first arrival of B...
  24. W

    I Poisson distribution with conditional probability

    Hi guys, I have a question about computing conditional probabilities of a Poisson distribution. Say we have a Poisson distribution P(X = x) = e^(−λ)(λx)/(x!) where X is some event. My question is how would we compute P(X > x1 | X > x2), or more specifically P(X> x1 ∩ X > x2) with x1 > x2? I...
  25. U

    Can Poisson Distribution Help with Predicting Multiple Outcomes in 1x2 Wagers?

    please am new here and i need a help, i can use poisson distribution to get a probability but how can i use it to get several outcome? like over and under ,1x2? thanks
  26. 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 \}...
  27. A

    I Proof to the Expression of Poisson Distribution

    Hello. Given a range of time in which an event can occur an indefinite number of times, we say a random variable X folows a poisson distribution when it follows this statements: X is the number of times an event occurs in an interval and X can take values 0, 1, 2, … The occurrence of one event...
  28. mr.tea

    How Does the Poisson Summation Formula Apply to Uniformly Convergent Series?

    Homework Statement let ##g## be a ##C^1## function such that the two series ##\sum_{-\infty}^{\infty} g(x+2n\pi)## and ##\sum_{n=-\infty}^{\infty} g'(x+2n\pi)## are uniformly convergent in the interval ##0\leq x \leq 2\pi ##. Show the Poisson summation formula: ##\sum_{n=-\infty}^{\infty}...
  29. chi_rho

    I Why do we require conditions for the Poisson Distribution?

    Three conditions must be met in order for the Poisson Distribution to be used: 1) The average count rate is constant over time 2) The counts occurring are independent 3) The probability of 2 or more counts occurring in the interval $n$ is zero Simply, why must these conditions be met for valid...
  30. V

    I Poisson Equation Neumann boundaries singularity

    I am trying to solve the poisson equation with neumann BC's in a 2D cartesian geometry as part of a Navier-Stokes solver routine and was hoping for some help. I am using a fast Fourier transform in the x direction and a finite difference scheme in the y. This means the poisson equation becomes...
  31. F

    A How to solve the following Poisson bracket

    anyone can help me how to solve the following poisson bracket? {U(x,λ), U(y,µ)} = −(1/λµ) {S^{i} (x), S^{j} (y)} σ_{i} ⊗ σ_{j} where U(x, λ) = −(i/λ) S(x)
  32. B

    I How can I verify the solution for Poisson equation with forgotten factor?

    Hello I have a physics article who solve Poisson equation of the form: L2 ∂2f/∂x2=sinh(f) The proposed solution is: tanh(f/4)=exp(x/L) tanh(f0/4) with f0 a constant I suspect an error, something like a forgotten factor. How can I verify? (I tried but I failed) I forgot the limit...
  33. I

    Normal approximation to Poisson random variable

    Homework Statement Suppose that the number of asbestos particles in a sam- ple of 1 squared centimeter of dust is a Poisson random variable with a mean of 1000. What is the probability that 10 squared cen- timeters of dust contains more than 10,000 particles? Homework Equations E(aX+b) =...
  34. M

    A Uncertainties in Poisson processes

    Good morning PF, I'm feeling a bit doubtful about this issue. I'm working with optical detectors and I have to characterize them in terms of quantum efficiency and other similar things. Now suppose my detector is, ideally, a single large pixel, which I illuminate for a specific time. Then I...
  35. ChrisVer

    I Fit a Poisson on Gaussian distributed data

    Hi, I have a simple/fast question... Can you reliably use a Poisson function to fit on data that seem to be Gaussian distributed (although that is due to the large number of the mean)?
  36. pizzicato

    A discrete equivalent of the Poisson coefficient

    Hello, I'm about elaborating a discrete mass-spring model to describe the vibration of a thin isotropic plate. For the flexion i choose a kind of spiral spring in the two directions X and Y: so the momentums will be Mx= Cbx.(Delta Thêta) ; My = Cby.(Dela Psi). and the energies: Eb =...
  37. 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...
  38. S

    Non-zero Poisson Distribution: Estimating Parameters and Confidence Intervals

    Homework Statement Non-Zero Poisson Probability distributions Scenario: The following data represents the number of typographical errors (typos) a lecturer makes (per page) in a document they are producing. A document's pages was inspected and the number of typos per page was recorded...
  39. Dakarai

    Show that the Poisson Mass Function has the same value for λ and λ-1

    Homework Statement Show that, quite generally, the Poisson Mass Function has the same value at λ (i.e. the average) and at (λ-1). Homework Equations The Poisson Mass Function is p(x) = [e^(-λ) * λ^(x)]/(x!) The Attempt at a Solution I started out by plugging in λ-1 into the equation to get...
  40. C

    Calculating Variance of Y using Poisson and Binomial Distributions

    Homework Statement I need to find the variance of Y. The number of errors on a page follows a Poisson distribution with lambda = 0.40 average . Y = the number of pages without error among the first 112 pages . Homework EquationsThe Attempt at a Solution In Poisson, I know that Variance =...
  41. A

    How Can We Calculate Band Bending Using Schrödinger-Poisson Theory?

    I am reading a lot about how to calculate band bending from solving the Schrödinger equation and Poisson equation self-consistently. To recap some of the central ideas are: We look at the conduction band of some semiconductor. If we assume that the electrons are free electrons with some...
  42. E

    Poisson brackets commutator vs. quantum commtation relation

    If we have Poisson bracket for two dynamical variables u and v, we can write as it is known ... This is for classical mechanics. If we write commutation relation, for instance, for location and momentum, we obtain Heisenberg uncertainty relation. But, what is a pedagogical transfer from...
  43. Rampart

    Poisson Process-theory questions

    Hey guys, what's up? I have some questions regarding the Poisson Process. I checked some threads, but not all, so forgive me if these questions have been answered before. 1)Lets say I am given some some tables or/and graphs or generally speaking some data and I am asked to find out if the...
  44. 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?
  45. S

    Rotation transformation by poisson brackets

    Homework Statement Can anybody suggest hints on how to show that x'=xcosΘ-ysinΘ, y'=xsinΘ+ycosΘ by using the infinite string of poisson brackets? Homework Equations ω→ω+a{ω,p}+a^2/2!{{ω,p},p}+... The Attempt at a Solution Sorry, I just can’t think of any way, substituting doesn’t work.
  46. A

    Poisson, Einstein, Weak Energy Condition

    Hello In Newtonian theory Poisson's equation holds: ## \nabla ^{2} U = 4 \pi G \rho ##. So: given a density ##\rho ##, it is possible to find a potential U. On the other hand, I can choose a random function U and give it a gravitational significance if it gives, by Poisson's eq., a density...
  47. Y

    MHB What Is the Probability of 30 Car Accidents in Eight Random Days?

    Hello all, I have this Poisson distribution question, which I find slightly tricky, and I'll explain why. The number of car accidents in a city has a Poisson distribution. In March the number was 150, in April 120, in May 110 and in June 120. Eight days are being chosen by random, not...
  48. 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...
  49. A

    Poisson Distrib.: Estimating Mean of Data Set

    Homework Statement I am given a data set known to come from a poisson distribution. Homework Equations Poisson distribution The Attempt at a Solution I want to calculate the mean of the data set for use in the Poisson Distribution function. How do I best estimate this. Do I take the...
  50. A

    What is the closed form expression for the sum: Σx2/n! from 0 to N?

    Is there a closed form expression for the sum: Σx2/n! from 0 to N for N=∞ it is an exponential but what about the finite case?
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