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. 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...
  2. G

    How Is the MGF of Zt Derived in a Poisson Process?

    Homework Statement Suppose cars arriving at an intersection follow a poisson distribution and that the number of passengers in the nth car is Yn where n>=1 and they are iid and independent of Nt each with moment generating function My(k) Write Zt as a sum of iid random variables and show...
  3. S

    Probability of event occurring - poisson distribution?

    probability of event occurring -- poisson distribution? I am the keeper of records for my local Volunteer Fire Dept. I have now collected data for each of our incident calls from the last 3 years and have made some _very_ basic stabs at interesting statistics which you can see at...
  4. Z

    How to Calculate Crime Probability Using Poisson and Binomial Distributions?

    Hey guys, I'm kind of stuck on this question. In a certain town, crimes occur at a Poisson rate of 2.4 per month (i.e. according to a Poisson process with a rate of 2.4 per month). What is the probability of having exactly 2 months (not necessarily consecutive) with exactly 4 crimes during...
  5. snoopies622

    Poisson Brackets: A Simple Example in Classical Mechanics

    Could someone show me a simple example of the usefulness of Poisson brackets - for instance, a problem in classical mechanics? I know the mathematical definition of the Poisson bracket, but from there the subject quickly seems to get very abstract.
  6. O

    Solving fluid's Poisson equation for periodic problem or more easy way?

    The problem is about mathematics but it originates from the self-gravitational instability of incompressible fluid, so let me explain the situation first. I have an incompressible uniform fluid disk that is infinite in the x-y direction. The disk has a finite thickness 2a along the...
  7. T

    Help with Poisson equation in irregular mesh

    Hi all, I've written a program that solves the Poisson equation in an irregular mesh, using a finite volume discretization. the method works well, and the solution is very good at a distance of 2-3 mesh sizes off of the source. The problem is that I need to know what is the contribution of...
  8. T

    Calculating Poisson Probability for Car Rental Income

    Homework Statement A car rental shop has four cars to be rented out on a daily basis at $ 50 per car. The average daily demand for cars is four. (1) Calculate the expected daily income received from the rentals (2) If the shop wishes to have one more car, the additional cost incurred...
  9. Y

    Help with Poisson problem in a unit ball.

    I cannot get the answer of the problem as in the book but the book usually right. I did it in 2 totally different ways and I still get my own answer. Can anyone help me double check? This is to find u(r,\theta,\phi) given: \nabla^2 u(r,\theta,\phi) = -k u(r,\theta,\phi) = f(r)=1 \hbox {...
  10. F

    CDF of the ratio of Poisson and possibly-Poisson R.V.

    Hello, This is my first post - so let me know if I communicate incorrectly. To start, note that my thread title may be misleading as to my actual problem. I think it describes my situation, but let me provide background and then restate my problem as I see it, so as to allow for a...
  11. K

    The quadratic covariation of Brownian motion and poisson process

    Hi: I want to know the quadratic covariation of Brownian motion B(t) and poisson process N(t).Is it B(t)? Thanks !
  12. Saladsamurai

    Probability Poisson Process and Gamma Distribution

    Homework Statement The Attempt at a Solution Part (a) is no problem, it is simply P(X>10) = 1 - P(X<=10) which requires the use of tabulated cumulative poisson values. Part (b) is throwing for a loop. I know that I need to invoke the Gamma distribution since that is what the...
  13. Y

    I don't agree with the solution manual of a mixed poisson problem

    Homework Statement Solve mixed poisson's problem on disk given \nabla^2 U= r sin \theta \hbox{ for } 0 <r< \frac{1}{2} \nabla^2 U= 0 \hbox{ for } \frac{1}{2} < r < 1 With given boundary condition U(1,\theta)=0 2. Answer from the solution manual U(r,\theta) =...
  14. A

    Poisson Process Conditonal Probabilities

    Hey I'm really struggling with this: What is the expected value of a poisson process (rate λ, time t) given that at least one even has occured? I was told the best way was to find the conditional distribution first. So this is: P(Xt=z | Xt≥1) = P(Xt=z, Xt≥1) / (PXt≥1) = P(Xt=z) /...
  15. T

    Poisson process with different arrival rates

    Homework Statement I cannot figure out this example: suppose that initially individuals enter a room from one door according to a Poisson process with arrival rate lambda1. Suppose that as soon as one inidividual enters, this door is shut down and a second door is open. The numer of...
  16. V

    Calc Poisson Bracket: {π,∂φ} Calculation

    How can I work out {π,∂φ} where {,} is a Poisson Bracket; π is the canonical momentum and ∂φ is the spatial derivative of the field (ie. not including the temporal one). Basically the question boils down to (or atleast I think it does!), working out ∂(∂φ) /∂φ - ie. differentiating the...
  17. D

    Random Processes | Poisson or not? | Probability of doing n jobs in t hours

    Homework Statement The number of hours that it takes to process a certain type of job is a random variable with mean and standard deviation 2. AAssuming that processing times are independent, approximate the probability that atleast 50 jobs can be sequentially processed within 240 hours...
  18. D

    Stochastic Processes, Poisson Process | Expected value of a sum of functions.

    Homework Statement Suppose that passengers arrive at a train terminal according to a poisson process with rate "$". The train dispatches at a time t. Find the expected sum of the waiting times of all those that enter the train. Homework Equations F[X(t+s)-X(s)=n]=((($t)^n)/n!)e^(-$t))...
  19. A

    Stiff- One-Dimensional Poisson Equation in Plasma

    Hey Guys, So I am trying to model the development of a collisional plasma in time. Now the problem I face is at the sheath boundary the changes in the charge densities is very large. I use the charge densities to evaluate the electric potential at different points in the plasma. I have...
  20. R

    No Poisson Ratio for Link Element in FEA

    in finite element analysis there is no poisson ratio for LINK ELEMENT (truss structure) just explain ?
  21. L

    Statistics Question: The 3rd Moment of Poisson Distribution

    Homework Statement X is a discrete random variable that has a Poisson Distribution with parameter L. Hence, the discrete mass function is f(x) = L^{x} e^{-L} / x!. Where L is a real constant, e is the exponential symbol and x! is x factorial. Without using generating functions, what is...
  22. Y

    Solving 2D Poisson problem with a single series

    Solving 2D Poisson problem with a single series! Conventional solution of \nambla^2u(x,y)=f(x,y) involve solution u(x,y)= \sum_{n=1}^{\infty} \sum_{m=1}^{\infty}E_{mn} sin(\frac{m\pi}{a}x) sin(\frac{n\pi}{a}y) This is a two series solution which is tedious to solve. The book PDE by Asmar...
  23. S

    Probability Question About The Poisson Probability Distribution

    Probability Question About "The Poisson Probability Distribution" Homework Statement - Assume that 1 in 200 people carry the defective gene that causes inherited colon cancer. A sample of 1000 individuals is taken. Use the Poisson approximation to calculate the appoximate standard deviation...
  24. M

    Validating the Probability Function f(x) for Zero-Inflated Poisson Distribution

    Homework Statement f(x) = (1-p)+pe^-lamdba ; x=0 = [p(e^-lambda)lambda^x]/x! ; x = 1, 2, ... = 0 ; otherwise Homework Equations show that f(x) is a valid probability function The Attempt at a Solution I think I am supposed to integrate...
  25. R

    What is the Error in My Approach to the Poisson Process Problem?

    I'm a bit frustrated with this one... Let (X_t)_{t\geq 0} be a Poisson Process with rate \lambda Each time an 'arrival' happens, a counter detects the arrival with probability p and misses it with probability 1-p. What is the distribution of time, T until the first particle is detected? I...
  26. 8

    A poisson distribution question

    Homework Statement On average, each of the 18 hens in my henhouse lays 1 egg every 30 days. If I check the hens once per day and remove any eggs that have been laid, what is the average number, μ, of eggs that I find on my daily visits? What is the most probable (whole) number of eggs that I...
  27. E

    Poisson Probability Distribution

    Homework Statement Suppose that .10% of all computers of a certain type experience CPU failure during the warranty period. Consider a sample of 10,000 computers. a.)What are the expected value and standard deviation of the number of computers in the sample that have the defect? b.) What...
  28. D

    What is the Probability of 2 Events Occurring in a Poisson Process?

    Homework Statement Events X, Y, Z are all Poisson processes. Event X has a rate of 1 per unit time , event Y has a rate of 2 per unit time and event Z has a rate of 3 per unit time. Find the probability that 2 events (of any type) occur during the interval (0, 3). Homework Equations...
  29. R

    Probability of 1st Arrival From Poisson Process of Rate $\lambda$

    I did this question, but I'm unsure of my reasons behind it. I was hoping someone here could go through the problem for me. I got the answer 1/\lambda - 1/(\lambda + \mu). I did so by integrating, \int_0^\infty P(\text{one event from } \lambda \text{ in }(0, t]) \times P(\text{zero event...
  30. P

    Graph analysis - how closely histogram fits poisson curve

    Homework Statement Its about random radioactive decay. I have a histogram showing the number of counts recorded in 3 second intervals and I've drawn the Poisson Curve on the same graph. I have a graph for 50 intervals and one for 100 intervals and I need to analyse how well the data...
  31. P

    Graph analysis - how closely histogram fits poisson curve

    Homework Statement This isn't really a question, its a report thing. Its about random radioactive decay. I have a histogram showing the number of counts recorded in 3 second intervals and I've drawn the Poisson Curve on the same graph. I have a graph for 50 intervals and one for...
  32. P

    Poisson dist. with small numbers

    Dear Physicists, I have a poisson dist with a mean at 0.00107. I tried that usual SQRT(mean) for the standard deviation but of course I got an sigma larger than my actual plot. Can someone point me to some text or the right theory? Cheers L
  33. R

    Poisson Martingales and Gambler's Ruin

    I posted this in the HW help section, but I had no responses. I figure that this place may be better to answer this question. If this is against the rules or anything, mods please remove it! Homework Statement Find the probability of an outcome of Gambler's Ruin using a Poisson...
  34. R

    Poisson Martingales and Gambler's Ruin

    Homework Statement Find the probability of an outcome of Gambler's Ruin using a Poisson martingale. Let m be the parameter of a Poisson Process (ie the lambda) Let N(t) be a continuous Poisson process at time t>=0 Let M(t) = N(t) - mt Homework Equations Now, define s = inf{t | M(t) <=...
  35. R

    Quadratic Variation of a Poisson Process?

    Hey guys, This is my first post on PhysicsForums; my friend said that this was the best place to ask questions about math. Anyways, I have to find the Quadratic Variation of a Poisson Process. My professor doesn't have a class textbook (just some notes that he's found online), and...
  36. P

    Definition of Poisson Bracket: {f,g}

    Hi, what is the correct definition for a Poisson bracket? Some books say it is: {f,g} = df/dp.dg/dq - df/dq.dg/dp but others say it is: {f,g} = df/dq.dg/dp - df/dp.dg/dq One is the other multiplied by -1. Which is the correct definition? Thanks for any help.
  37. B

    Poisson Statistics in Solid State Physics

    Homework Statement In the Drude model the probability of an electron having a collision in an infinitesimal time interval dt is given by dt/\tau. (a) Show that an electron picked at random at a given moment will have no collisions during the next t seconds with probability e-t/\tau. (b) Show...
  38. K

    Poisson and binomial distributions, corrupted characters in a file

    A text file contains 1000 characters. When the file is sent by email from one machine to another, each character (independent of other characters) has probability 0.001 of being corrupted. Use a poisson random variable to estimate the probability that the file is transferred with no errors...
  39. P

    Probability of Poisson Distribution: Nr of Customers in Shop

    Nr of customers arriving at a shop follow Poisson. In 15, an average of 4 customers arrive. a) A customer has just arrived. Then a minute passed and no one arrived. What is the probability of it takoing at least 5 more min. until another customer arrives? b) Consider 40 non-overlapping...
  40. N

    Estimating the Mean for a Batch of 50 Items Using the Poisson Distribution

    A machine on average produces 4 defective items out of a batch of 100 items. Find the probability that a batch of 50 items has 3 defective items in it using the Poisson probability distribution. the problem is.. i just want to know the mean or average value for batch of 50 items.. i got...
  41. K

    Poisson process: compute E[N(3) |N(2),N(1)]

    note: N(t) is the number of points in [0,t] and N(t1,t2] is the number of points in (t1,t2]. Let {N(t): t≥0} be a Poisson process of rate 1. Evaluate E[N(3) |N(2),N(1)]. If the question were E[N(3) |N(2)], then I have some idea... E[N(3) |N(2)] =E[N(2)+N(2,3] |N(2)] =E[N(2)|N(2)] +...
  42. H

    Proof of Normal approximation to Poisson.

    I have been looking for a proof of the fact that for a large parameter lambda, the Poisson distribution tends to a Normal distribution. I know the classic proof using the Central Limit Theorem, but I need a simpler one using just limits and the corresponding probability density functions. I was...
  43. E

    Negative binomial, Poisson, or gambler's ruin?

    Homework Statement . Peter and Paul bet one dollar each on each game. Each is willing to allow the other unlimited credit. Use a calculator to make a table showing, to four decimal places, for each of p = 1/10, 1/3, .49, .499, .501, .51, 2/3, 9/10 the probabilities that Peter is...
  44. E

    Poisson distribution (would you please verify?)

    Homework Statement 1. Suppose that the number of telephone calls an operator receives from 9:00 to 9:05 A.M. follows a Poisson distribution with mean 3. Find the probability that the operator will receive: a. no calls in that interval tomorrow. b. three or more calls in that interval the...
  45. N

    Proving Poisson Brackets Homework Statement

    Homework Statement f(p(t),q(t)) = f_o + \frac{t^1}{1!}\{H,f_o\}+\frac{t^2}{2!}\{H,\{H,f_o\}}+... Prove the above equality. p & q are just coords and momenta How do we do this if we don't know what H is? Where do we start? Homework Equations The Attempt at a Solution
  46. D

    How to Determine Scalar Potential Inside and Outside a Charged Sphere?

    Homework Statement Use Poisson's equation and Laplace's equation to determine the scalar potential inside and outside a sphere of constant charge density po. Use Coulomb's law to give the limit at very large r, and an argument from symmetry to give the value of E at r=0. Homework...
  47. R

    MATLAB Solving Poisson Equation with FFT in MATLAB

    Hi, As part of my homework, I wrote a MatLab code to solve a Poisson equation Uxx +Uyy = F(x,y) with periodic boundary condition in the Y direction and Neumann boundary condition in the X direction. I used the finite difference method in the X direction and FFT in the Y direction to...
  48. R

    Poisson Distribution and Chebyshev's Inequality

    Homework Statement LEt X have a Poisson distribution with u=100. Use Chebyshev's inequality to determine a lower bound for P(75<x<125) Homework Equations Chebyshev's Inequality. The Attempt at a Solution I'm really unsure of how to go about calculating this problem. Any help...
  49. M

    Independent poisson random variables

    Homework Statement There are two urns, A and B. Let v be a random number of balls. Each of these balls is put to urn A with probably p and to urn B with probability q = 1 - p. Let v_a and v_b denote the numbers of balls in A and B, respectively. Show that random variables v_a and v_b are...
  50. Q

    Using Poisson Approximation to Compare Infection Rates in Village A and B

    There are 60 infections in village A per month and 48 infections in village B per month. Let A be no of infections in village A per month and B be no of infections in village B per month. Assume occurrence is independent and random. So Method 1 (Working method): A~Po (60) and B~Po (48)...
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