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. I

    A Numerical solution of the Poisson equation

    In my case, there is proton radiation acting on the material. Consequently, the protons get stuck in the sample and create an electrostatic field. I would like to solve the Poisson equation inside the sample. I assume that the medium is infinite and homogeneous, that is, the potential at...
  2. D

    Using a neural network to solve the Poisson equation

    To train the model, I generated a set of deterministic solutions with random boundary conditions ##u(-1)=a## and ##u(1)=b##. I then added a small amount of noise to these solutions. However, the model's accuracy is significantly worse compared to the most basic finite difference methods. Is...
  3. D

    Poisson process is identical on equal intervals?

    Let ##N_t## be the Poisson point process with the probability of the random variable ##N_t## being equal to ##x## is given by $$\frac{(\lambda t)^xe^{-\lambda t}}{x!}.$$ ##N_t## has stationary and independent increments, so for any ##\alpha\geq 0, t\geq 0,## the distribution of ##X_t =...
  4. F

    A Poisson noise on ##a_{\ell m}## complex number: real or complex?

    1) In a cosmology context, when I add a centered Poisson noise on ##a_{\ell m}## and I take the definition of a ##C_{\ell}## this way : ##C_{\ell}=\dfrac{1}{2\ell+1} \sum_{m=-\ell}^{+\ell} \left(a_{\ell m}+\bar{a}_{\ell m}^{p}\right)\left(a_{\ell m}+\bar{a}_{\ell m}^{p}\right)^* ## Is Poisson...
  5. L

    Conserved quantities via Poisson brackets

    Hi, Results from the previous task, which we may use I am unfortunately stuck with the following task Hi, I have first started to rewrite the Hamiltonian and the angular momentum from vector notation to scalar notation: $$H=\frac{1}{2m}\vec{p_1}^2+\frac{1}{2m}\vec{p_2}^2-\alpha|\vec{q_1}-...
  6. A

    Poisson random process problem

    Hello all, sorry for the large wall of text but I'm really trying to understanding a problem from a study guide. I am quite unsure on how to approach the following multi-part problem. Any help would be appreciated. Problem: Useful references I'm using to attempt the problem My attempt: For...
  7. Zuzana

    A LogLikelihood - Poisson distribution

    Hello :) I try to fit some parameters of the particle (e.g. energy, direction) be means of log-likelihood minimization. Input data to likelihood function are pulses amplitudes, while Poisson distribution is used. However, the problem is that Poisson distribution is as follows i.e. for higher...
  8. F

    A Relation between a_{\ell m} noise and Poisson noise with C_{\ell}

    We assume two galaxy population, ##\mathrm{A}## and ##\mathrm{B}##; the corresponding maps have the following ##a_{\ell m}## : ## \begin{aligned} &a_{\ell m}^{A}=b_{A} a_{\ell m}^{M}+a_{\ell m}^{p A} \\ &a_{\ell m}^{B}=b_{B} a_{\ell m}^{M}+a_{\ell m}^{p B} \end{aligned} ## Here, ##b_{A}## and...
  9. barryj

    Is My Understanding of Poisson Distribution Correct for Different Area Sizes?

    Please excuse me for posting in this group.l There seems to be little activity in the statistics group and i might get no response. For this example, lambda, is the average of dogs per 100 square miles = 0.05. if I wanted the probability of 2 dogs in 100 square miles I would calculate.. P(x=2)...
  10. K

    Using Poisson random variables to calculate this probability

    I calculated the mean which is 78.4 And the Standard deviation is 5.6 I thought the answer would be (90^(-78.4)/78.4!)*e^-90 But looking back having a decimal factorial doesn't make sense I have the numerical answers for c)= 0.019226 and d)=0.022750 but I my solution was wrong. Any help on...
  11. yucheng

    Confused about the nature of Laplace vs Poisson equation in BVP

    Hi! The problem clearly states that there is a surface charge density, which somehow gives rise to a potential. The author has solved the Laplace equation in cylindrical coordinates and applied the equation to the problem. So ##\nabla^2 V(r,\phi) = 0##, and ##V(a,\phi) = V_a(\phi)## (where...
  12. F

    A Calculating the variance of integrated Poisson noise on a defined quantity

    It is in cosmology context but actually, but it is also a mathematics/statistical issue. From spherical harmonics with Legendre deccomposition, I have the following definition of the standard deviation of a ##C_\ell## noised with a Poisson Noise ##N_p## : ## \begin{equation}...
  13. F

    A Photometric Galaxy Clustering Error and Poisson Noise

    The error on photometric galaxy clustering under the form of covariance which is actually a standard deviation expression for a fixed multipole ##\ell## : ## \sigma_{C, i j}^{A B}(\ell)=\Delta C_{i j}^{A B}(\ell)=\sqrt{\frac{2}{(2 \ell+1) f_{\mathrm{sky}} \Delta \ell}}\left[C_{i j}^{A...
  14. arcTomato

    I The power spectrum of Poisson noise

    I thought that if we Fourier transformed the counts of the sum of the signal from the source and the Poisson noise, and obtained the power spectrum, we would get the following, ##P_{j}=P_{j, \text { signal }}+P_{j, \text { noise }}+\text { cross terms }## but I found the following description...
  15. H

    I How to Find the Distribution of T for Competing Poisson Processes

    Hi all, I've been struggling for days now with this problem. Would appreciate any idea you might have. Red cars and blue cars arrive as independent Poisson processes on [0, ∞) with respective rates λ_r, λ_b. Let T denote the arrival time of the first red car whose nearest neighbor is a blue car...
  16. F

    A Analytical expression of Cosmic Variance - Poisson distribution?

    I have an expression of Matter Angular power spectrum which can be computed numerically by a simple rectangular integration method (see below). I make appear in this expression the spectroscopic bias ##b_{s p}^{2}## and the Cosmic variance ##N^{C}##. ## \begin{aligned}...
  17. A

    Poisson ratios for Orthotropic materials (composites)

    I'm new to composite materials. I've studied mechanical engineering but I am actually usually involved in hydrodynamics (in which I've done my masters). However for a project we do fluid structure interaction with composites, and as these things go, you cannot get away with the 'black box'...
  18. D

    Exploring the Structure of the Poisson Equation

    Theorem 1: There exists at most one solution ##u\in C^2(\bar{\Omega})## of the Dirichlet boundary value problem. Proof: (1) We assume there is a second solution ##\tilde{u}## of the Dirichlet boundary value problem. We compute $$\Delta v=\Delta (u-\tilde{u})\Rightarrow -\Delta u + \Delta...
  19. D

    Solving the Poisson equation with spherically symmetric functions

    I tried to follow the method outlined in lectures, and ended up with an incorrect solution. My understanding of PDEs is a bit shaky so I thank anyone for constructive feedback or information. :bow: The solution to the Poisson equation \begin{equation} -\Delta u(x)=\frac{q}{\pi...
  20. Addez123

    B Poisson distribution having variation coefficient = .5?

    Variation coefficient is calculated by And the very definition of poisson distribution is that $$\mu = \sigma $$ So how would any other value but 1 be a possible?
  21. B

    Poisson Ratio -- Finding a corresponding analytical solution for the strain

    Hi, I ran into problems using the poisson ratio. For a FE simulation I created a simple 2D 1mm x 1mm block, and prescribed a 0.1 mm displacement at the top edge. Furthermore, the bottom edge is constraint in the y-dir, and the left edge in the x-dir. The material parameters are E = 100, and v...
  22. arcTomato

    A Average of the power spectrum of Poisson noise

    I am learning about noise that follows a Poisson distribution. When I do a Fourier transform of the data with only Poisson noise to get the power spectrum, what is the average value of the power spectrum?
  23. arcTomato

    A Why the Poisson noise level is set to 2 after applying Leahy norm

    I am studying about power spectrum analysis in high energy astrophysics. I cannot understand why the Poisson noise level is set to 2 after applying Leahy normalization. $$P_{j}=2 /_{N \mathrm{ph}}\left|a_{j}\right|^{2}$$ The above is the equation for leahy norm, Can I expand the equation from...
  24. A

    I Finite Lorentz Transformation via Poisson Bracket

    Let me define ##L_{x;v}## as the operator that produce a Lorentz boost in the ##x##-direction with a speed of ##v##. This operator acts on the components of the 4-position as follows $$L_{x;v}(x) =\gamma_{v}(x-vt),$$ $$L_{x;v}(y) =y,$$ $$L_{x;v}(z) =z,$$ $$L_{x;v}(t)...
  25. E

    Young's Modulus and the Poisson Ratio of Aluminium

    Hi. I've been given these sets of value. How do I calculate Young's modulus and Poisson Ratio from these set of value. I tried to create a stress vs strain graph but the graph does not look like a common stress vs strain graph but instead more of a y=x graph. The diameter of the rod is 10mm and...
  26. D

    Two Poisson distributed random variables

    How do I evaluate P(X-Y=0)=?
  27. jisbon

    Conditional Probability + Poisson Distribution

    Confused and not sure if it is correct, but please do correct my steps. We let event B be that there are at least 3 customers entering in 5 minutes. Hence P(B) = 1- P(X=0)- P(X=1) - P(X=2) = ##1- \dfrac{e^{-5}5^{0}}{0!}-\dfrac{e^{-5}5^{1}}{1!}-\dfrac{e^{-5}5^{2}}{2!} ## = 0.8753... Now we let...
  28. L

    Solving the 1D Poisson equation for a MOS device

    Hey everyone, I'm currently working on a 1D Poisson Solver for a MOS device (Al-Si-SiO2). Therefore, I programmed a Poisson Solver which is appling a boxintegration (Finite Volume Method) through the structure from φ(0) at the metal-oxide interface and φ(x_bulk = 20 nm) in in the silicon bulk...
  29. Luke Tan

    I Invariance of the Poisson Bracket

    I've recently been starting to get really confused with the meaning of equality in multivariable calculus in general. When we say that the poisson bracket is invariant under a canonical transformation ##q, p \rightarrow Q,P##, what does it actually mean? If the poisson bracket ##[u,v]_{q,p}##...
  30. domingoleung

    B Poisson Distribution - Selecting cookies that are indistinguishable

    Here's the problem: A chef made 500 cookies randomly mixed with 1000 nuts including 600 almonds and 400 hazelnuts in which each nut is the same size. Suppose the number of pieces of nuts in a piece of cookie follows a Poisson distribution. (a) Suppose cookies are randomly selected one-by-one...
  31. D

    Maths for Sound passing through different mediums

    What is the mathematics involved with calculating the energy lost from sound as it passes through different mediums? If I started off with a 70dB(A) sound wave, and after 0.5m it passed through 10mm of mild steel - what would be the sound level (in dB) 1m away from the steel plate? To clarify...
  32. D

    Solving the 3D Poisson equation

    Hello ! I want to solve the 3D Poisson equation using spherical coordinates and spherical harmonics. First I must solve this : ##d^2\phi/dr^2 + 1/rd\phi/dr - l*(l+1)/r^2 = \rho (r)## with ##\phi (\infty ) = 0## (here ##\phi## is the gravitationnal potential and ##\rho## is the mass density)...
  33. Daniel Petka

    B Poisson Spot Madness: Laser & Coin Lens Experiment

    Recently, I started to experiment with a laser and a coin used as a lens, being inspired by an old Cody's Lab video. My initial assumption was that through diffraction, the laser will be focused onto a spot on when the coin is a certain distance away from the wall. In a way, I imagined it as an...
  34. arcTomato

    How can I effectively suppress Poisson noise in my data?

    Hi all I would like to know how to suppress the spectrum of Poisson noise. At first, I tried "binning". I made the data of Poisson noise which sums up 4Hz sin wave.(the data number##N=10000##,and the data time is 1s) and I average out the data every 100bins. After this, I derived the power...
  35. arcTomato

    The power spectrum of Poisson noise

    Dear all. I have made the power spectrum of Poisson noise(expected value ##λ=2##), and it becomes like this I think this is not good. I don't know why the power is so high when Random variable(x axis) is 0. I tried another expected value version ,but the result didn't change. so I would like to...
  36. M

    I Poisson Error Q: Can I Add Errors in Quadrature?

    Hello! I have a fit to a histogram ##y(x)##. Now I want to predict the number of counts at some other point, not in the original data, using this fitted function and assign an error to it. Let's say that the the point where I want to compute this value, ##x_0## gives ##y(x_0) = 100##. As the...
  37. F

    Poisson distribution and Shot Noise

    My setup: I have the an LED (LED370E) in front of a photodiode (S12915-16R). The photodiode is connected to an ADC (DT5751) which has a counting functionality. The way it works is that it counts how many times the signal goes above a certain threshold and makes a histogram out of it. I know...
  38. SunilS

    A Relationship between Poisson distribution and Poisson Process

    Apologies if this has been discussed elsewhere. I know a Poisson process implies a Poisson distribution, but does a Poisson distribution imply a Poisson process? and does the absence of a Poisson distribution imply the absence of a Poisson process? TIA - Sunil
  39. S

    Poisson distribution probability problem

    So I thought you would find the probability of having 0 errors when the mean rate is 1.6. Square that by 5 and multiply that by one minus the probability of having 0 errors to the power of 7. So that is basically the probability of having 0 errors to the power of 5 multiplied by the probability...
  40. K

    Calculate potential form poisson equation

    Hi. I've the following charge density: ## \rho = \rho_0 \frac {r}{R} ## I'm getting a trouble to calculate the potential inside a sphere of radius R located in the center of axis with the given charge density (using poisson equation): the Laplacian in spherical coordinates is: ##\frac {1}{r^2}...
  41. user366312

    I Difference between Time, Arrival-Time & Inter-Arrival-Time in Poisson Process

    . The above are some of the typical problems related to Poisson Process. I need to understand the difference between time, inter-arrival time, and arrival time in this regard. Say, we start our counting from 9:00 AM and count up to 10:00 AM. Image-1: arrival process. 1. 1st call comes at...
  42. user366312

    A Deriving a Probability Generating Function for Independent Poisson Variables

    Can anyone kindly tell me how I can derive a Probability Generating Function of Poisson Distribution for ##X+Y## where ##X## and ##Y## are independent? I know that PGF for a single variate Poisson Distribution is: ##G(t) = e^{-\lambda (1-t)}##. Then how can I derive a PGF for the same? Is...
  43. user366312

    A Question about the Poisson distributed of random variable

    The following is related to Poisson process: $$P(N_1=2, N_4=6) = P(N_1=2, N_4-N_1=4) = P(N_1=2) \cdot P(N_3=4)$$ Why is $$(N_3=4)=(N_4-N_1=4)$$? Can anyone explain?
  44. A

    MHB Best procedure to determine Lambda to calculate Poisson probability

    What is the best procedure to determine Lambda to calculate the Poisson probability? Say I want to calculate P(X ≥1) of an accident occurring next day. For this I would calculate the average of daily accidents and divide it by 10. The question is, should I take the previous 10 days? Or calculate...
  45. MrsTesla

    Variance with Poisson distribution

    <Moderator's note: Moved from a technical forum and thus no template.> So, I have this problem and I am stuck on a sum. The problem I was given is the following: The probability of a given number n of events (0 ≤ n < ∞) in a counting experiment per time (e.g. radioactive decay events per...
  46. A

    I Uncertainty of MonteCarlo Simulations: Weight and Error Bars

    Hello everybody, I need a help, primarly a confirmation about my reasoning. I have data from a MonteCarlo simulation of collisions between particles at LHC (made with Madgraph). I have plotted some variables, for example the angle between two final leptons. Then I have normalized the plot to a...
  47. T

    Conditional Expectations of 2 Variables

    Homework Statement Suppose that the number of eggs laid by a certain insect has a Poisson distribution with mean ##\lambda##. The probability that anyone egg hatches is ##p##. Assume that the eggs hatch independently of one another. Find the expected value of ##Y##, the total number of eggs...
  48. Z

    Bernoulli, Binomial & Poisson: What is pi?

    Homework Statement Hi, I have a confusion in knowing Pi in the equations attached. Eq are related to the Topic Discrete Random Variables in the context of Probability lecture I also can't understand what is P(Y=y|Pi)? Homework Equations Eq are attached The Attempt at a Solution I can't...
  49. Livio Arshavin Leiva

    A What exactly is a "rare event"? (Poisson point process)

    These days I've been reading in the internet about the Poisson Distribution because that was a concept I couldn't manage to understand completely when I studied it, so since then I've been always quite curious about Poisson processes, and how there are a lot of natural phenomena (mostly the...
  50. S

    Poisson Distribution -- Rental of a number of television sets

    Homework Statement A dealer has a stock of 6 similar television sets which he rents out to customers on a monthly basis. It is known from past experience of the dealer that the monthly demand for the television sets have a Poisson distribution with mean 3.56 (i) Find the probability that in any...
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