Poisson Definition and 508 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. B

    Water Poisson Ratio: What & How?

    hello everybody. can we define a poisson ratio for fluids, eg for water?? if no, why? n if yes, how?? thnx.
  2. R

    Poisson distribution normalized

    how can i show that the Poisson distribution is properly normalized?
  3. S

    Poisson Distribution problem in probability for engineers

    In 2003, there were many media reports about the number of shark attacks. At the end of the year, there were a total of 30 unprovoked shark attacks. By comparison, there were 246 shark attacks over the prior ten years. (a) Give an expression for the probability of more than 30 shark attacks...
  4. M

    Stochastic processes: infinite server queue with batch poisson arrivals

    Hi everyone, I am trying to solve this problem but I am stuck with doubts. Here are my ideas. Homework Statement Busloads of customers arrive at an infinite server queue at a Poisson rate λ Let G denote the service distribution. A bus contains j customers with probability aj = 1...
  5. E

    Proving the poisson distribution is normalized

    SOLVED Homework Statement I am trying to prove that the poisson distribution is normalized, I think I've got an ok start but just having trouble with the next step. Homework Equations A counting experiment where the probability of observing n events (0≤n<∞) is given by...
  6. R

    Poisson distribution for insurance

    Homework Statement An insurer uses the Poisson distribution with mean 4 as the model for the number of warranty claims per month on a particular product. Each warranty claim results in a payment of 2 by the insurer. Find the probability that the total payment by the insurer in a given...
  7. H

    Solving Poisson Distributions for Expected Values of Sales

    Homework Statement A store selling newspapers orders n = 4 of a certain newspaper because the manager does not get many calls for that publication. If the number of requests per day follows a Poisson distribution with mean 3, what is the expected value of the number sold? Homework Equations...
  8. C

    Deriving the Distribution of Total Time in a Poisson Process Series

    Homework Statement Imagine you want to go from A to C via B. So you have two steps: A to B and B to C. Let's assume the time taken (t1) to go from A to B is Poisson and is given by Pab(t1) and the for B to C is t2 and the distribution is Pbc(t2). You are given: Pab(t1) = k1exp(-k1t1) and...
  9. C

    How to Derive the Distribution of Two Poisson Processes in Series?

    Homework Statement Imagine you want to go from A to C via B. So you have two steps: A to B and B to C. Let's assume the time taken (t1) to go from A to B is Poisson and is given by Pab(t1) and the for B to C is t2 and the distribution is Pbc(t2). You are given: Pab(t1) = k1exp(-k1t1) and...
  10. P

    Poisson Distribution/ uniform dist.

    I am in an error analysis class and our homework has asked us this (we will be writing a computer program to do this): "Create a sample with 396 draws from a Poisson distribution with N=1000 and 4 draws from the uniform distribution between 0 and 105. This sample represents data from a CCD...
  11. J

    Calculating E(T) for Poisson Processes with Intensity a

    Homework Statement Let N(t) be a Poisson process with intensity a. Let T be the occurrence of the first event in (0,t] for some t>0, if there is such a one. If there are no events in (0,t] we set T=t. Compute E(T). Homework Equations None. The Attempt at a Solution I have no clue...
  12. S

    Steel - Youngs Modulus, Poisson Ratio

    Hello, Quick question on how you would go about calculating this. A grade of steel has the following properties; Tensile strength = 300 N/mm^2 - (not relevant?) Youngs Modulus = 200 GPa Poisson Ratio = 0.3 The grade of steel is 2m long, with a 20mm cross sectional area. It is positioned...
  13. fluidistic

    Poisson brackets little problem

    Homework Statement For a particle, calculate Poisson brackets formed by: 1)The Cartesian components of the linear momentum \vec p and the angular momentum [/itex]\vec M =\vec r \times \vec p[/itex]. 2)The Cartesian components of the angular momentum.Homework Equations [u,p]_{q,p}= \sum _k...
  14. Rasalhague

    Poisson & normal distributions as approximations for the binomial

    These three quotes talk about the use of the Poisson and normal distributions as approximations for the binomial when n is large. The first two quotes here say Poisson is best when p small, and the normal otherwise. The third seems to change the story; it says Poisson is best for large p too. Is...
  15. B

    Olbers' paradox - Poisson model

    Olbers' paradox states that if the universe is infinite, static and homogeneous then why is the night sky dark. Of course it's been resolved but it brings up an interesting probability question: If we model the universe with a spatial Poisson model (probability that a small element is...
  16. M

    Poisson probability distribution

    Homework Statement Homework Statement A particle detector is set up to detect type A particles. These are detected as a poisson process with parameter lamda = 0.5 per day. (i) What is the probability that 3 or more will be detected in anyone day? (ii) What is the distribution of...
  17. P

    Poisson inflow, constant outflow question

    The following problem came up in my work. You have a tube, open at the top. Raindrops fall into the mouth of the tube at a mean rate i per second, 0 <= i < 1, in a Poisson process. There's a hole in the bottom of the tube. When there's water in the tube, it flows out at a constant rate of 1...
  18. A

    Calculating Probability in Poisson Process Problem | Z(t-c)=m, Z(t)=k

    Given a poisson process Z(t) with a given rate lamda, k and m nonnegative integers and t and c real and positive numbers, calculate the probability: P(Z(t-c)=m | Z(t)=k) thanks
  19. G

    Poisson distribution problem help

    I need help with aPoisson distribution problem please. Question is: company capable of handling 5 calls every 10 min on new system. Prior to new system, company analysts determined incoming calls to the system are Poisson distributed w/ a mean equal to 2 every 10 min. what is the probability...
  20. B

    Poisson Brackets / Levi-Civita Expansion

    Hi, I am stumped by how to expand/prove the following identity, \{L_i ,L_j\}=\epsilon_{ijk} L_k I am feeling that my knowledge on how to manipulate the Levi-Civita is not up to scratch. Am i correct in assuming, L_i=\epsilon_{ijk} r_j p_k L_j=\epsilon_{jki} r_k p_i Which...
  21. M

    Characteristic Function of a Compound Poisson Process

    Hello, I am trying to find a characteristic function (CF) of a Compound Poisson Process (CPP) and I am stuck :(. I have a CPP defined as X(t) = SIGMA[from j=1 to Nt]{Yj}. Yj's are independent and are Normally distributed. So, in trying to find the CF of X I do the following: (Notation...
  22. F

    Poisson distribution-solve for x

    Homework Statement How to find for a Poisson distribution the number of successes for a given probability and mean. For example, for mean, \lambda, of 1, and a required probability of 0.01, what would the number of successes in the time interval be?Homework Equations...
  23. B

    Calculating Poisson ratio is a way to measure a material's response to stress.

    Homework Statement question A ciclindrical metal specimen 10 mm in diameter is stressed elastically in tension. A force of 15000 N produces a reduction in diameter of 0.007 mm. Compute Poisson ratio if its elastic modulus is 100 GPa Homework Equations E=stress/strain The...
  24. B

    How to Compute Poisson Ratio Correctly?

    question A ciclindrical metal specimen 10 mm in diameter is stressed elastically in tension. A force of 15000 N produces a reduction in diameter of 0.007 mm. Compute Poisson ratio if its elastic modulus is 100 GPa my attempt D original = 0.01m change in D = 0.000007m F=15000N E=...
  25. S

    Predictive distribution of poisson

    [b]1. The distribution of flaws along the length of an artificial fibre follows a Poisson process, and the number of flaws in a length L is Po(L\theta ). Very little is known about \theta. The number of flaws in five fibres of lengths 10, 15, 25, 30 and 40 metres were found to be 3, 2, 7, 6, 10...
  26. U

    Canonical Transformations, Poisson Brackets

    This isn't actually a homework problem, but a problem from a book, but as it's quite like a homework problem I thought this forum was probably the best place for it. Homework Statement Consider a system with one degree of freedom, described by the Hamiltonian formulation of classical...
  27. Oxymoron

    Poisson PDF with non-integer support

    Homework Statement If X is a Poisson random variable with \lambda = 2 find the probability that X>0.5. Homework Equations The Poisson PDF: P(x,\lambda) = \frac{\lambda^k}{k!}e^{-\lambda} The Attempt at a Solution Usually with these sorts of probability problems where they ask...
  28. D

    Probability theory - Poisson and Geometric Random Variable questions

    Homework Statement [/b] There are two problems I need help with, which are posted below. Any help is appreciated. 1)Let X have a Poisson distribution with parameter λ. If we know that P(X = 1|X ≤ 1) = 0.8, then what is the expectation and variance of X? 2)A random variable X is a sum of...
  29. S

    Most powerful test involving Poisson

    Homework Statement The number of sales made by a used car salesman, per day, is a Poisson random variable with parameter \lambda. Given a random sample of the number of sales he made on n days, what is the most powerful test of the hypothesis Ho: p = 0.10 versus Ha: p = 0.25, where p is the...
  30. T

    Difference between Renewal Process and Poisson Processes

    Hey All, Can someone please explain to me the difference between a Poisson Process and a Renewal Process ? is it just that the Holding times for Poisson processes are exponential and Holding times for Renewal Processes are any kind of probability distribution (as the wiki page seems to imply)...
  31. S

    What is the Probability Distribution of Parking Requests in a Poisson Process?

    The number of cars driving past a parking area in a one-minute time interval has a Poisson distribution with mean lambda. The probability that any individual driver actually wants to park his or her car is p. Assume that individuals decide whether to park independently of one another. a)If one...
  32. M

    Solving Poisson Distribution Homework for 50 Liters of Sediment

    Homework Statement In one of the archaeological excavation sites, the artifact density (number of prehistoric artifacts per 10 liters of sediment) was 1.0. Suppose you are going to dig up and examine 50 liters of sediment at this site. Let r = 0, 1, 2, 3,… be a random variable that represents...
  33. R

    Probability - Poisson Random Variable

    Homework Statement A trial consists of tossing 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...
  34. B

    Poisson Distribution and slot machine

    Homework Statement A casino slot machine costs C dollars per play. On each play, it generates random variable X ~ Poisson with parameter λ < 1, and pays the player X! (X factorial) dollars. As a function of the fixed parameters λ and C, how much money would you expect to win (or lose) per turn...
  35. H

    Gaussian Elimination Solution to the 2D Poisson Equation

    I am trying to use Gaussian elimination to solve the 2D poisson equation. I've done this for the 1D problem without problems, but for some reason my solution for the 2D problem is incorrect; it looks something like the correct solution but it's as if the resulting field were cut in half, so...
  36. N

    Understanding Poisson Distribution: Explanation & Examples

    The explanation for the Poisson distribution in reference book is " when given an interval of real number, assume events occur at random throughout the interval. If the interval can be partitioned into subintervals of small enough length such that 1. the probability of more than 1 event in a...
  37. F

    Two independent Poisson processes (one discrete, one continuous)

    Hi Guys, I've used this forum as a great resource for a while now and it's always helped me out. Now I'm really stuck on something and was hoping you guys could help out. It's a pretty long question, but if you guys can just give me a general direction of what to do, I can go ahead and work it...
  38. G

    Solve for the covariance in the bivariate Poisson distribution

    Dear All, The bivariate Poisson distribution is as follows, \[ f(y_{s},y_{t})=e^{-(\theta_{s} + \theta_{t}+\theta_{st})}\frac{\theta_{s}^{y_{s}}}{y_{s}!}\frac{\theta_{t}^{y_{t}}}{y_{t}!} \sum_{k=0}^{min(y_{s},y_{t})} \binom{y_{s}}{k} \binom{y_{t}}{k}...
  39. R

    Poisson equation: what shape gives largest area average

    When considering the solution u(x,y) of the poisson equation u_xx + u_yy = -1 for (x,y) in G on a 2-dimensional domain G with Dirichlet boundary conditions u = 0 for (x,y) on boundary of G I am wondering the following: for what shape of the domain G do I obtain the largest area-average...
  40. P

    Solving a Probability Question with Poisson Distribution

    Homework Statement I am solving a particular probability question using Poission distributin after the Solving I get an equation Homework Equations 1.67 = e^-a (1/1-a) I ought to get the value of a from the equation but I was Unable to go further from here . The Attempt at a...
  41. T

    Suppose X and Y are independent Poisson random variables,

    Suppose X and Y are independent Poisson random variables, each with mean 1, obtain i) P(X+Y)=4 ii)E[(X+Y)^2] I m trying to solve this problem but have difficulty starting ... If some one could give me a some pointers
  42. T

    If X and Y are independent poisson variates

    Question: If X and Y are independent poisson variates with mean λ1 and λ2 respectively, what is the probability that i) X + Y =k ii) X = Y Solution: Dont know how to solve this .
  43. T

    Poisson errors for the distribution of galaxies?

    Poisson errors for the distribution of galaxies?? Hi, I have some data regarding the distribution of galaxies of varying mass in different density regions of the Universe, from which I have a mass functions for each region. I would now like to introduce some errors so I can determine whether...
  44. Q

    Proof of Poisson: Proving P_t Not in {0,1}

    How can I prove that P( there exist a t>0 : the changeP_t is not in {0,1} ) = 0 where (P_t)_t>0 is a piosson Proces with parameter lambda > 0. Thank you.
  45. M

    Practical use of binomial and Poisson Distribution in the field of engineering

    Hi... Hope i 'll get the good result that where we practically use the binomial and poisson distribution in the field of engineering...
  46. K

    How do I evaluate this Poisson distribution?

    Homework Statement How do I evaluate this Poisson distribution? Homework Equations The Attempt at a Solution So I have figured out what the values for lambda and x are, but I don't know how to evaluate once I plug the values into the formula. λ = 20 x = 18 = [ e^-20...
  47. T

    Probability question; Conditional probability and poisson distribution

    Homework Statement A radioactive source emits particles according to a Poisson process, at an average rate of λ per unit time. Each particle emitted has probability p of being detected by an instrument, independently of other particles. Let X be the number of particlese emitted in a given...
  48. F

    Independence in Poisson Process

    I'm studying the Poisson Process (rate R) and I'm hung up on the issue of dependence. This seems like and easy question but I have no background in probability whatsoever. By definition, the number of events in disjunction time intervals are independent. Okay. Fine. But say we have an...
  49. R

    Poisson approximation to the normal

    So my book merely mentions that this holds as a result of the central limit theorem for values of lambda greater than 10, but ideally greater than 32. Anyway I was wondering if anyone knew this actual proof as I am interested in seeing it step by step and I could not have found it anywhere...
  50. R

    Poisson Distribution Homework: Use Central Limit Theorem

    Homework Statement We can approximate a poisson distribution from the normal. Suppose lambda is a large positive value; let X ~ Poisson(lambda) and let X1...Xn be independant identicly distributed from a Poisson (lambda/n) distribution. Then X and X1+...+Xn have the same distribution. Use the...
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