What is Poisson distribution: Definition and 138 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. 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...
  2. 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...
  3. 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...
  4. 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).
  5. 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...
  6. 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...
  7. 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) =...
  8. 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...
  9. 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...
  10. 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...
  11. 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∞...
  12. 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...
  13. 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...
  14. 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...
  15. 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...
  16. 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...
  17. 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...
  18. 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...
  19. 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...
  20. 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.
  21. S

    How Does Poisson Distribution Affect the Efficacy of a Cold-Reducing Drug?

    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...
  22. T

    How Do You Calculate One Sigma Confidence Intervals for Poisson Events?

    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...
  23. A

    Poisson distribution with efficiency problem

    Hi, I have a problem with determining the probability distribution function of the number n of detector counts in a given time t. I am assuming the events follow exponential distribution ε(t,λ) = λexp(-λt). Now if that was everything it would simply be a Poisson distribution, however, what I...
  24. B

    Poisson distribution problem

    The number of cracks in a section of interstate highway that are significant enough to require repair is assumed to follow a Poisson distribution with a mean of two cracks per mile. What is the probability that there are no cracks that require repair in 5 miles of highway? any help guys? :)
  25. P

    MHB Poisson Distribution: Prob of <=3 Wrong Connections in Building

    A building has 2 independent automatc telephone exchanges A and B. The number X of wrong connections for A in anyone day is a poisson variable with parameter 0.5 and the number Y of wrong connections for B in any one day is a poisson variable with parameter 1. Calculate in any particular...
  26. S

    Poisson Distribution - why are these different?

    Homework Statement X(t) is a Poisson process with \lambda=0.2 events per second. What is the probability of zero events in 45 seconds? 2. The attempt at a solution \frac{45}{0.2}=225 (0.2 second intervals) so P[X=0] in 225 consecutive intervals is: \left(e^{-0.2}\right)^{225} = 2.86...
  27. M

    How to find equations for confidence limits in Poisson distribution?

    Homework Statement What kind of equations you'll get when trying to find confidence limits 100(1-a) % for λ in Poisson distribution? Homework Equations Poisson distribution P(X=x) = e-λ λx / x! (x=0,1,2 ...) The Attempt at a Solution I made an equation as follows: Ʃ (k = from k0 to n) e-λ...
  28. X

    Calculating Poisson Distribution for Telephone Calls in College Switchboard

    Homework Statement Telephone calls enter a college switchboard according to a Poisson process on the average of three calls every 4 minutes (i.e., at a rate of λ=0.75 per minute). Let W denote the waiting time in minutes until the second call. Compute P(W>1.5 minutes). Homework Equations...
  29. Y

    Bio-statistics, Poisson distribution

    I'm studying Bio-statistics and I came across this problem from the textbook.It's actually answered on the back of the book, but I couldn't really get the same numbers. i Desease-free infants at the end of month i 0 2500 1 2425 2 2375 3 2300 4 2180 5 2000 6 1875 7 1700 8 1500 9...
  30. A

    How to Get Covariance of Bivariate Poisson Distribution

    Dear all, I have a problem in solving covariance of Bivariate Poisson Distribution Let X_i \sim POI (\theta_i) , i = 1,2,3 Consider X = X_1 + X_3 Y = X_2 + X_3 Then the joint probability function given : P(X = x, Y = y) = e^{\theta_1+\theta_2+\theta_3} \frac {\theta_1^x}{x!} \frac...
  31. B

    Poisson Distribution of Accidents

    Homework Statement In New York in the last 3 years there were 55 driving accidents. Assume all days are alike. What is the approximate probability that "in the next 3 years there will be at least 2 days with more than one accident". Homework Equations Poisson approximation The...
  32. N

    Why does the Poisson distribution apply here?

    Hello, I'm reading a text about statistics, but I don't understand why Poisson applies. (Note, this is not an assignment or anything like that.) Why would X be Poisson distributed with that parameter theta? The only Poisson that I could find reasonable is modelling X as Poisson...
  33. T

    Calculate Posterior p.f. from Poisson Distribution

    Homework Statement Suppose that the number of defects on a roll of magnetic recording tape has a Poisson distribution for which the mean λ is either 1.0 or 1.5, and the prior of λ is the following: L(1.0)=0.4 and L(1.5)=0.6 If the roll of tape selected at random is found to have 3...
  34. J

    Poisson Distribution for Drill Stock Management

    Homework Statement A hardwhere store sells on average 8 drills per week. The store receives ONE delivery of drills at the same time each week. Find the no. of drills that need to be in stock after a delivery for there to be at most a 5% chance of the store NOT having sufficent drills to meet...
  35. P

    Poisson distribution and random processes

    Hello! I am writing because I recently became interested in probability distributions, and I have to you a few questions. Poisson distribution is given as a probability: f(k;\lambda)=\frac{\lambda^{k}e^{-\lambda}}{k!} But what is lambda? Suppose that we consider as an unrelated incident...
  36. C

    Drude Theory of Metals Poisson Distribution Problem

    Homework Statement This is the first problem from Ashcroft's Solid-State Physics which I recently picked up due to having far too much free time. The first two parts of the problem relate to the probability that an electron picked at random will have had no collision during the preceding t...
  37. R

    Poisson distribution normalized

    how can i show that the Poisson distribution is properly normalized?
  38. 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...
  39. 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...
  40. X

    Statistics question (poisson distribution, multivariate )

    Homework Statement On average, 2.5 telephone calls per minute are received at a corporation's switchboard. Making appropriate assumptions about the distribution ( provide justification ), find the probability that at any given minute there will be more than 2 calls. Homework Equations No...
  41. 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...
  42. 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...
  43. 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...
  44. 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...
  45. 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...
  46. 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...
  47. 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}...
  48. 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...
  49. 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...
  50. 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...
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