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. R

    Poisson equation with finite difference method

    Hi guys , i am solving this equation by Finite difference method. (dt2/dx2 + dt2/dy2 )= -Q(x,y) i have developed a program on this to calculate the maximum temperature, when i change the mesh size the maximum temperature is also changing, Should the maximum temperature change with mesh...
  2. S

    Solving Poisson Distribution: Part IV - Tank of Water (10^5 cm3)

    Homework Statement i am having problem with part iv ) . the ans is 0.04519 . can anyone tell me how to do this ? i have solved part i , ii and iii ..p/s line 1: A tank contain 10^5 cm3 of water Homework Equations The Attempt at a Solution
  3. Z

    Canonical Transformation / Poisson Brackets

    Question: (A) Show that the following transformation is a canonical transformation: Q = p + aq P = (p - aq)/(2a) (B) Find a generating functions for this transformation. Attempt of Solution: Alright, so this seems to be a very straight forward problem. Transformations are canonical...
  4. S

    Poisson distribution involving 2 variables

    Homework Statement A coffee shop sell tea and coffee. The number of cups of coffee sold in a minute can be assumed to be a random poisson variable with mean = 1.5 . The number of cups of tea sold can be assumed to be an independent random variable with mean = 0.5. Calculate the probablity...
  5. 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?
  6. Greg Bernhardt

    How Do Poisson Brackets Function in Classical Mechanics?

    Definition/Summary In the Hamiltonian formulation of classical mechanics, equations of motion can be expressed very conveniently using Poisson brackets. They are also useful for expressing constraints on changed canonical variables. They are also related to commutators of operators in...
  7. M

    MHB Help with Poisson Distribution

    Hi guys I got a question on the poisson distribution and never previously done stats at all. It follows: The mean count of a radioactive substance is 25 disintegrations per minute. Using the Poisson distribution, find the probability that, in a time of 12 seconds, there are- i) No...
  8. T

    Probability through Poisson Distributon

    Hi! I am aware of the steps used to show that (e^-λ*λ^r)/r! is P(X=r) for X~Po(λ), where λ = E(X) = Var(X). I have two questions regarding this: - I'm aware that all of the probabilities add up to 1, but how do we know that they're all probabilities and not just a set of values that add to 1...
  9. A

    Solving for Poisson Probability Change w/ Function x

    Suppose that a system is such that in a time dt, the probability that an event A occurs, given that it has not already happened, is given by: P(t,t+dt) = w(t) * dt The solution for the probability that A has occurred at a time t is something like: P(t) = 1 - exp(∫0tw('t)dt') Now...
  10. T

    Geometric, Exponential and Poisson Distributions - How did they arise?

    I'm going through the Degroot book on probability and statistics for the Nth time and I always have trouble 'getting it'. I guess I would feel much better if I understood how the various distribution arose to begin with rather than being presented with them in all there dryness without context...
  11. S

    Poisson distribution (radioactive decay)

    Homework Statement I am a freshman in physics, just done a lab about radioactive decay. I've measured the # of beta particles per second 400 times and got the frequency of each number K using Excel. I'm supposed to take the data and fit it to a puason distribution in MATlab. The data points...
  12. Z

    Solve 3D Poisson & Laplace Equations w/ Non-Zero BCs

    Hello every one and thank you in advance, I'm try to solve 3D Poisson equation analytically not numerically, but the help i found has the boundary conditions equal to zero, there is anyone to have a step by step process to solve Poisson and/ or Laplace 3D equation where the boundary conditions...
  13. R

    MHB Boundary integral method to solve poisson equation

    Suggest how to solve Poisson equation \begin{equation} σ ∇^2 V = - I δ(x-x_s) δ(y-y_s) δ(z-z_s) \nonumber \end{equation} by using the boundary integration method to calculate the potential $V(r,z)$ with the help of changing the Poisson equation into cylindrical polar co ordinates? Where V is...
  14. 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...
  15. U

    Probability of Sample Mean for Poisson Distribution

    Homework Statement A rectangular field is gridded into squares of side 1m. at one time of the year the number of snails in the field can be modeled by a Poisson distribution with mean 2.25 per m^2. (i) a random sample of 120 squares is observed and the number of snails in each square...
  16. M

    Weak Form of the Poisson Problem

    Hi, I know the weak form of the Poisson problem \nabla^2 \phi = -f looks like \int \nabla \phi \cdot \nabla v = \int f v for all suitable v. Is there a similarly well-known form for the slightly more complicated poisson problem? \nabla (\psi \nabla \phi ) = -f I am writing some finite...
  17. R

    Hamiltons equations of motion in terms of poisson bracket

    In Hamiltonian formulation there is an expression df / dt = { f , H } + ∂f / ∂t where f is function of q, p and t. While expressing Hamiltons equations of motion in terms of Poisson Bracket, i.e if the function f = q of p then its partial time derivative ∂f / ∂t becomes zero.. Please explain why?
  18. 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...
  19. S

    MHB Find p(x=0 or 1) & F(x) for Poisson Distribution

    If p(x=1)=p(x=2) where x follows a Poisson distribution, then find p(x=0 ~~or~~ 1) . Also find F(x)In connection with the above question, I have confusion about the last part i.e., about F(x). I can find E(x) here, but how to find F(x).
  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. S

    Probability: Poisson distribution involving customer arrivals

    Homework Statement There are two stores A and B. Customers can equally enter one of the two stores, i.e., for a specific customer, the probabilities she enters store A or B both are 0.5. If the total number of customers in two stores has the Poisson distribution of parameter λ, then...
  22. T

    Poisson distribution on a simulated (SSA) data set

    I've been asked to fit the histogram with a Poisson distribution as part of a mostly independent learning thing. The data was produced through a stochastic simulation. Can someone get me started on how I would go about finding the expected distribution? If you need additional information...
  23. DocZaius

    Poisson vs Binomial approaches yield different results

    Homework Statement I made this question for myself to try to see if I could use two approaches (Poisson Distribution and Binomial Distribution) to solve a problem: Someone's average is to make 1 out of every 3 basketball shots. What are the chances she makes exactly 2 shots in a trial of 3...
  24. 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]) >...
  25. maistral

    Poisson PDE discretisation help

    Okay, I'm trying to play around again :D A little overview; I know that the Poisson equation is supposed to be: uxx + uyy = f(x,y) I can manage to discretise the partial derivative terms by Taylor. I don't know how to deal with the f(x,y) though. Say for example, uxx + uyy = -exp(x). what...
  26. E

    Solve the screened Poisson equation

    Homework Statement Solve the equation \nabla^2\phi-\frac{1}{\lambda^2_D}\phi=-\frac{q_T}{\epsilon_0}\delta(r) substituting the \delta representation \delta(r)=\frac{1}{4\pi}\frac{q_T}{r} and writing the laplacian in spherical coordinates. Use as your guess...
  27. L

    Poisson Integral: Real vs Complex a

    ## \int^{\infty}_{-\infty}dxe^{-ax^2}=\sqrt{\frac{\pi}{a}}## Is it correct also when ##a## is complex?
  28. Y

    Is wave and heat equation with zero boundary Poisson Equation?

    I have two questions: (1)As the tittle, if u(a,\theta,t)=0, is \frac{\partial{u}}{\partial {t}}=\frac{\partial^2{u}}{\partial {r}^2}+\frac{1}{r}\frac{\partial{u}}{\partial {r}}+\frac{1}{r^2}\frac{\partial^2{u}}{\partial {\theta}^2} and \frac{\partial^2{u}}{\partial...
  29. Y

    Question on using solution from Helmholtz equation in Poisson equation

    Helmholtz equation stated that \nabla^2 u(r,\theta,\phi) =-ku(r,\theta,\phi) = f(r,\theta,\phi) This is being used for Poisson equation with zero boundary: \nabla^2 u(r,\theta,\phi) = f(r,\theta,\phi) and u(a,\theta,\phi)=0 I just don't see how this can work as k=m^2 is a number only...
  30. V

    Preservation of Poisson Bracket Structure upon quantization?

    When (canonically) quantizing a classical system we promote the Poisson brackets to (anti-)commutators. Now I was wondering how much of Poisson bracket structure is preserved. For example for a classical (continuous) system we have $$ \lbrace \phi(z), f(\Pi(y)) \rbrace = \frac{\delta...
  31. E

    Proving Some Poisson Bracket identities - a notational question

    Proving Some Poisson Bracket identities -- a notational question I need some help just understanding notation, and while this might count as elementary it has to do with Hamiltonians and Lagrangians, so I posted this here. Homework Statement Prove the following properties of Poisson's...
  32. darida

    Verifying a Canonical Transformation with Poisson Brackets

    Homework Statement Show that Q_{1}=\frac{1}{\sqrt{2}}(q_{1}+\frac{p_{2}}{mω}) Q_{2}=\frac{1}{\sqrt{2}}(q_{1}-\frac{p_{2}}{mω}) P_{1}=\frac{1}{\sqrt{2}}(p_{1}-mωq_{2}) P_{2}=\frac{1}{\sqrt{2}}(p_{1}+mωq_{2}) (where mω is a constant) is a canonical transformation by Poisson bracket test. This...
  33. A

    Poisson Probability: At Least 50% Defective Brake Lights

    Homework Statement Suppose that 1% of cars have defective brake lights and n cars are to be inspected. How large should n be for the sample to have a probability of at least 50% of containing a car with a defective brake light? Give an answer using a Poisson approximation with an appropriate...
  34. 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...
  35. K

    MHB Verification of poisson approximation to hypergeometric distribution

    How can I verify that $\lim_{N,M,K \to \infty, \frac{M}{N} \to 0, \frac{KM}{N} \to \lambda} \frac{\binom{M}{x}\binom{N-M}{K-x}}{\binom{N}{K}} = \frac{\lambda^x}{x!}e^{-\lambda}$, **without** using **Stirling's formula** or the **Poisson approximation to the Binomial**? I have been stuck on...
  36. Y

    Is Helmholtz equation a Poisson Equation?

    Helmholtz equation:##\nabla^2 u=-ku## is the same form of ##\nabla^2 u=f##. So is helmholtz equation a form of Poisson Equation?
  37. StevieTNZ

    Poisson Statistics + Photon Detections

    Hi there, Having done a Google, I wasn't able to find much information relating specifically to Poisson statistics and photon detections. I was wondering why photon detection experiments are calculated using Poisson statistics? (So for example, would Poisson distribution calculations be...
  38. Darth Frodo

    Poisson distribution. E(X) not a whole number.

    Homework Statement During the day, cars pass along a point on a remote road at an average rate of one per 20 minutes. Calculate the probability that; (i) in the course of an hour no car passes; (ii) in the course of 30 minutes exactly 4 cars pass;Homework Equations P(X = x) =...
  39. D

    What is the Probability of Insect Contamination in Multiple Chocolate Bars?

    Homework Statement Data from www.centralhudsonlab.com determined the mean number of insect fragments in 225-gram chocolate bars was 14.4, but three brands had insect contamination more than twice the average. Assume the number of fragments (contaminants) follows a Poisson distribution...
  40. L

    MHB How to derive the Poisson p.m.f.

    Can anyone derive the p.m.f. of Poisson distribution without mentioning the binomial distribution? The binomial deriving method put lambda = np and finally the binomial p.m.f. become the Poisson one as n goes to infinity. It seems that this is only proving that binomial distribution will...
  41. C

    Poisson distribution & exponential decay

    Homework Statement t(s) = 1 15 30 45 60 75 90 105 120 135 N(counts) = 106 80 98 75 74 73 49 38 37 22 Consider a decaying radioactive source whose activity is measured at intervals of 15 seconds. the total counts during each period are given. What is...
  42. C

    Properties of Poisson Distribution

    Homework Statement Use the Poisson distribution W=(λ^n/n!)*e^-λ to calculate <n> Homework Equations <n>=ƩW*n The Attempt at a Solution Since W = (λ^n/n!)*e^-λ I wind up with <n>=[(λ^n/n!)*e^-λ]*n But I really don't know where to go from here. Should I do a Taylor Series. I've...
  43. M

    Probability Theory: Poisson Distribution

    Homework Statement A random variable has a Poisson distribution with parameter λ = 2. Compute the following probabilities, giving an exact answer and a decimal approximation. P(X ≥ 4) Homework Equations P(X = k) = λke-λ/k! The Attempt at a Solution P(X ≥ 4) = Ʃk = 4∞...
  44. 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...
  45. L

    Prove convergence in probability for n * Poisson variable to zero

    The problem: Let \mu_{n} = \frac{1}{n} for n \in \mathbb{N}. Let X_{n} \; \mathtt{\sim} \; \textrm{ Poisson}\left( \lambda_{n} \right). Let Y_{n} = n X_{n}. Show that Y_{n} \xrightarrow{P} 0 . Work I've done: I've shown that X_{n} \xrightarrow{P} 0 by showing that \mathbb{P} \left(...
  46. J

    How Much Stock Should a Grocer Keep to Avoid Running Out with Less Than 1% Risk?

    Homework Statement On the average, a grocer sells 4 of a certain article per week. How many of these should he have in stock so that the chance of his running of stock within a week will be less than 0.01? Assume Poisson distribution. Homework Equations The Attempt at a...
  47. L

    Poisson again - totally stumped as compressive column?

    Homework Statement Column supports a mass on its' top. So force is downwards. Column properties: Do = 50mm (outer dia) Di = 40mm (inner dia) E = 250 GNm^-2 (modulus of elasticity) V = 0.33 (Poissons ratio) Homework Equations Poissons ratio = Transverse strain = - εt / εl Transverse strain...
  48. J

    Poisson Distribution w/ book errors

    Homework Statement In a lengthy manuscript, it is discovered that only 14% of the pages contain no typing errors. If we assume that the number of errors per page is a random variable with a Poisson distribution, find the percentage of pages that have: Exactly one typing error, At the most 2...
  49. N

    Poisson bracket and Electric and Magnetic Weyl tensor in GR

    In order to understand how related are the theories of General Relativity and Electromagnetism, I am looking at the electric and magnetic parts of the Weyl tensor (in the ADM formalism) and compare them with the ones from Maxwell's theory. I have tried to look at the Poisson bracket, but the...
  50. F

    Neumann and Dirichlet BCs in discrete Poisson EQ

    Hello all. I am working on a problem and I am getting a bit confused. Suppose we have a poisson equation that we wish to solve subject to certain boundary conditions. Let's say we are in 1D (we can later extrapolate to more dimensions). Is it possible to impose Dirichlet boundary...
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