Poisson Definition and 441 Threads

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...

Hi, Suppose you have a collection of fields \phi^i (t,x) depending on time and on 1 space variable, for i=1,...,N. Its dynamics is defined by the Lagrangian L=\frac{1}{2} g_{ij}(\phi) (\dot{\phi}^i \dot{\phi}^j - \phi ' ^i \phi ' ^j ) + b_{ij}(\phi) \dot{\phi}^i \phi ' ^j where...

Since I couldn't find any reference on the subject of Poisson bracket formalism of classical field theory, I'm posting a few question here: A) What are the Poisson brackets of the source-less EM field? B) Does the law that the Poisson brackets between a dynamical variable and its conjugate...

I want to find [M_i, M_j] Poisson brackets. $$[M_i, M_j]=\sum_l (\frac{\partial M_i}{\partial q_l}\frac{\partial M_j}{\partial p_l}-\frac{\partial M_i}{\partial p_l}\frac{\partial M_j}{\partial q_l})$$ I know that: $$M_i=\epsilon _{ijk} q_j p_k$$ $$M_j=\epsilon _{jnm} q_n p_m$$ and so...

Hi, Referring to Jackson's Electrodynamics 3ed, page 197, line 5. He assumes that the magnetization can be divided into volume part and surface part, thus generating eqn 5.100. This is fine. In a straightforward way, I wanted to do the same but for electrostatics, eqn 4.32:∅= (1/4πε) ∫dv...

Homework Statement Solve the equation: ∂2F/∂x2 + ∂2F/∂y2 = f(x,y) Boundary Conditions: F=Fo for x=0 F=0 for x=a ∂F/∂y=0 for y=0 and y=b Homework Equations How can I find Eigengunctions of F(x,y) for expansion along Y in terms of X? The Attempt at a Solution I can't imagine...

Homework Statement Consider ##n## independent trials, each of which results in one of the outcomes ##1,...k## with respective probabilities ##p_1,...p_k, \sum_{i=1}^{k} p_i = 1##. (I interpret this summation as just saying mathematically that definitely one of the outcomes has to occur on each...

Why Poisson kernel is significant in mathematics? Poisson kernel is ##P_r(\theta)=\frac{1-r^2}{1-2rcos\theta+r^2}##. http://www.math.umn.edu/~olver/pd_/gf.pdf page 218, picture 6.15. If we have some function for example ##e^x,sinx,cosx## what we get if we multiply that function with Poisson...

Dear all I am working with a Gamma-Poisson mixture distribution where (and this is not usual) the support of the Gamma distribution (in fact, I am able to restrict to the neg exponential instead of the Gamma...) is finite, e.g. [0,1]. I would like to derive the mean and a Likelihood...

Homework Statement Let X_1,...,X_n be a random sample from a poisson distribution with mean \lambda Find the MLE of \lambda^2 + 1 Homework Equations The Attempt at a Solution I found \hat{\lambda}=\bar{x} Can I just square it and add 1 and solve for lambda hat? If not I have no idea...

Hello, hopefully this is the right place. This is a homework question, so it should definitely be in this forum, but I wasn't sure which sub-forum to put this rather elementary stats question. Homework Statement In my introductory mathematical statistics class, we've been given the...

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...

Let Y|X be a Poisson(X), and X be Gamma(\alpha, \beta). Find E(X|Y)... Since Y|X is Poisson(X), we have f(Y|X)= \frac{m^x e^{-m}}{x!}... Since X is Gamma(\alpha, \beta), we have f(x)= \frac{x^{\alpha-1} e^{-x/B}}{\Gamma(\alpha) \beta^{\alpha}}... Since f(Y|X) = \frac{f(x,y)}{f(x)} ====>...

Okay there is a particular equation in my book, which I just can't seem to understand intuitively. I've been staring at it for an hour now without progress, so I hope some of you can explain it. Basically it's the one on the attached picture. Let me introduce the notation so you can help me...

Homework Statement Homework Equations The Attempt at a Solution Here's what I've tried so far, not really sure how to go on with these problems. Been reading the textbook up and down on Poisson processes! Any hints or help? Especially 1.B, 1.C and 2.B, 2.C

What is the significance of the Poisson kernel (besides solving the Dirichlet problem)? What is the Poisson's role in solving the Dirichlet problem? I know it is the solution but what is meant by its role?

Prove: $$ \lim_{r\to 1}P(r,\theta) = \begin{cases} \infty, & \theta = 0\\ 0, & \text{otherwise} \end{cases} $$ For the first piece, take the summation $$ P(1,0) = \frac{1}{\pi}\left(\frac{1}{2} + \sum_{n = 1}^{\infty} 1^n\right). $$ Then $\sum\limits_{n = 1}^{\infty} 1^n = \infty$. Therefore, we...

For a fixed $r$ with $0\leq r < 1$, prove that $P(r,\theta)$ is an even function.Take $-r$. Then \begin{alignat*}{3} P(-r,\theta) & = & \frac{1}{2\pi}\frac{1 - (-r)^2}{1 - 2(-r)\cos\theta + (-r)^2}\\ & = & \frac{1}{2\pi}\frac{1 - r^2}{1 + 2r\cos\theta + r^2} \end{alignat*} I have $1 +...

Homework Statement Consider the Klein-Gordan action. Show that the Noether charges of the Poincare Group generate the Poincare Algebra in the Poisson brackets. There will be 10 generators.Homework Equations \{ A,B \}=\frac{\delta A}{\delta \phi}\frac{\delta B}{\delta \pi}-\frac{\delta...

Hi! I'n my quantum mechanics homework I've been asked to proved the Poisson summation formula. The mathematicians seem to use abstract and confusing notation when proving this kind of thing so I'm hoping for some help from physicists in standard notation ;) I'm starting with a function f(x) =...

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...

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.

Homework Statement Let f(q, p), g(q, p) and h(q, p) be three functions in phase space. Let Lk = εlmkqlpm be the kth component of the angular momentum. (i) Define the Poisson bracket [f, g]. (ii) Show [fg, h] = f[g, h] + [f, h]g. (iii) Find [qj , Lk], expressing your answer in terms of...

Hey guys, In class, I was shown that the Binomial prob density function converges to the Poisson prob density function. But why does this show that the Binomial distribution converges in distribution to the Poisson dist. ? Convergence in distribution requires that the cumulative density...

Hi friends I am trying to drive constraints of a Lagrangian density by Dirac Hamiltonian method. But I encountered a problem with calculating one type of Poisson Bracket: {\varphi,\partial_x\pi}=? where \pi is conjugate momentum of \varphi. I do not know for this type Poisson Bracket I can...

Hello everybody! I am using in my studies this beautiful book by Kippenhahn & Weigert, "Stellar Structure and Evolution", but I have some problems about collapsing polytropes (chapter 19.11)... After defining dimensionless lenght-scale z by: r=a(t)z and a velocity potential \psi...

Suppose a person takes data (say counts per minute of cars going past his window), for a long time. Then he loses his data, but knows that he counted 5 cars more often than any other number. What is the likely range for the average count rate? I tried to solve this by saying the mode is 5, so...

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...

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...

Hi, I am working on finding a solution to Poisson equation through Green's function in both 2D and 3D. For the equation: \nabla^2 D = f, in 3D the solution is: D(\mathbf x) = \frac{1}{4\pi} \int_V \frac{f(\mathbf x')}{|\mathbf x - \mathbf x'|} d\mathbf{x}', and in 2D the solution is: D(\mathbf...

Hi, I'm getting some confusing results and can't figure out what is wrong Suppose we have a uniform field E=[0,0,E_z] in a dielectric media. By E=-\nabla\psi we can deduce that \psi(x,y,z)=-z E_z But, taking the Laplacian \nabla^2\psi=\frac{\partial^2 (-zE_z)}{\partial z^2}=0...

Homework Statement show that the integral of the poisson kernel (1-r^2)/(1-2rcos(x)+r^2) converges to 0 uniformly in x as r tend to 1 from the left ,on any closed subinterval of [-pi,pi] obtained by deleting a middle open interval (-a,a) Homework Equations the integral of poisson...

Solve: Δu=-1, u(1, theta)=sin(theta). 0<r<1, -pie<theta<pie, u finite at r=0 What I've done: u=u1+u2. Δu2=0, with u(1, theta)=sin(theta). So eventually u2=rsin(theta). The u1 problem however I am not sure how to solve. Eigenfunction expansion doesn't seem like it would work (though not...

Hello, If you have two observables f and g both of which start off as: f =0 and g =0 and you evaluate their possion bracket: {f,g}, will it necessarily be equal to zero? Also, if just f=0 and g wasn't zero, would {f,g} =0? Thanks!

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...

I need help from anyone urgently, I need C code for Solving Poisson Equation has known source with Neumann condition by using FDM (finite difference method) in 2D problem.

Hi Almost every text use as example poisson/exponential distribution to model clients arrival. What makes this distribution so good to fit in these cases? Please math arguments Regards

I'm struggling to understand Poisson brackets a little here... excerpt from some notes: I am, however, stumped on how this Poisson bracket has been computed. I presume ra and Aa(r) are my canonical coordinates, and I have \dot{r}_a = p_a - \frac{e}{c}A_a (r) with A_a = \frac{1}{2}\epsilon...

In book http://www.phy.uct.ac.za/people/horowitz/Teaching/lecturenotes.pdf in section 2 it is described transition from Poisson bracket into Canonical Commutation Relations. But it is written The experimentally observed phenomenon of incompatible measurements suggests that position and...

Let customers arrive according to a poisson process with parameter st and let $X_{t}$ denote number of customers in the system by time t. Consider an interval [t,t+h] with h small. Show that P(1 arrival)= sh + o[h], P(more than one arrival)=o[h] and P(no arrival)=1-sh+o[h]. I know P(1...

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? :)

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...

I want to show that ∇2ϕ=ρ/2, which governs gravity in Newtonian physics? I found solution of this question in [General Relativity for Mathematicians, R.K.Sachs and H.Wu, 1997, page 112&271]. Solution refer to optional exercise as follows: Let R^ be the (0, 4) –tensor field physically...

Homework Statement Suppose a book of 600 pages contains a total of 240 typographical errors. Develop a poisson approximation for the probability that three partiular successive pages are error-free. The Attempt at a Solution I say that the number of errors is poissondistributed...

Dear all, I wonder if anyone has come across this problem before and could point me to a relevant ref or tell me what terms I might search for: I am interested in a continuous time process in which two alternating events (call them A and B) occur. Each event has an exponentially...

I am trying to plot the flux from an Astrophysical source as a function of time. Due to the nature of the source, I am only receiving a handful of photons in each time bin. So imagine I had 10 observing periods of 10 days each, in which my telescope received the following number of photons...

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...

Homework Statement This is not really a homework just studying but I'm kinda stuck. So I am trying to find out how to formally write down the Charge Density for any distribution. Although I will not get into Green's Function or how to find V, I got that fine. My example will be a Rod of...

I'm having trouble doing a classic proof (integration par part and induction on r) for this : Pr(X>t)=Pr(Y ≤r−1), where X follows a gamma Γ(α = r, β = 1/λ) and Y a Poisson P (λt). Start with r = 1 (exponential distribution). I don't really understand what induction on r really means...

Hello, I'm looking at some sporting data (similar to goals in a match) and trying to figure out what distribution applies to their count per match. Typically, Poisson is used in the industry to model the distribution. When I look at the historical events, poisson isn't too bad, but tends...