Gaussian Definition and 706 Threads

Homework Statement The integral of (x^n)(e^(-a*x^2)) is easier to evaluate when n is odd. a) Evaluate ∫(x*e^(-a*x^2)*dx) (No computation allowed!) b) Evaluate the indefinite integral of x*e^(-a*x^2), using a simple substitution. c) Evaluate ∫(x*e^(-a*x^2)*dx) [from o to +inf] d)...

Hi! I was asked to smear an histogram with a gaussian smearing, but I don't understand what does it mean. Can you please explain me what is a gaussian smearing? Thank you

Hi everyone, I'm new here and this is my first post in this forum. ^^ Homework Statement Suppose that you observe a fluorescent object whose true location is x0. Individual photons come from this object with apparent locations xi in an approximately Gaussian distribution about x0...

1. A solid dielectric sphere of radius 10 cm has an electric charge uniformly distributed throughout its volume. The electric field at 5 cm from the center of the sphere is 8.6 x 10^4 N/C, pointing radially outward. Calculate the magnitude and direction of the electric field at a point 15 cm...

Consider a point charge q at the vertex of an arbitrary cube. If asked to consider the flux the cube experiences, q/8epsilon seems a natural answer, by constructing seven more such cubes to create an overall cube of 8 times the volume with q at its center. But, this doesn't make sense to me...

Suppose I have a Gaussian probability distribution: N_{A}(0,1). A set of values are generated from this distribution to which an arbitrary amount of Gaussian noise, say N_{B}(0,0.5), is added and then the N_{B} values sorted from lowest to highest. These are then digitised by assigning 0...

Good evening Im starting to learn quantum mechanics from Griffith's book however I am having problems when dealing with Gaussian integrals in the first chapter. What book should I read in order to understand this subject? are there resources about gaussian integrals out there? Thanks a lot.

Hello, I have a doubt about the distribution of random variables that maximize the differential entropy in a set of inequalities. It is well known that the Normal distribution maximizes the differential entropy. I have the following set of inequalities: T1 < I(V;Y1|U) T2 < I(U;Y2) T3 <...

I have been looking for a proof of correctness of Gaussian elimination, but alas, without much success. Most online resources explain how to apply the algorithm rather than proving correctness. That said, I have been looking for a proof to the following theorem, which is stated in Friedberg's...

I'm dealing with multivariate normal distributions, and I've run up against an integral I really don't know how to do. Given a random vector \vec x, and a covariance matrix \Sigma, how would you go about evaluating an expectation value of the form G=\int d^{n} x \left(\prod_{i=1}^{n}...

1 Why do we choose a spherical surface as gaussian surface for a point charge to calculate electric field? In my view, the reason may be i. If we take the point charge at centre, each point of spherical surface will be at same distance from the...

I tried to used the Crank-Nicolson method to solve the advection equation for a Gaussian pulse using the code on this page and I'm lost at trying to setup the initial values for the Gaussian which is passed as an array into the advection matrix. The Gaussian pulse is defined initial...

what changes does there occur in the result of the gaussian distribution "integration e^-alpha*x^2 dx=sqrt(pi/alpha) if i substitute that x^2 with some (x-a)^2? then what should be the integral result ?

Homework Statement consider this Gaussian distribution p(x)=Ae^-(a(x-b)^2) Homework Equations use integration p(x)dx=1 to find out the value of A The Attempt at a Solution hi, i know about the gaussian distribution formula integration e^-alpha*x^2 = sqrt(pi/alpha) now for...

I was looking to the definition of the Bell curve, and the Gaussian distribution, but I don't see any difference when we represent them in a graph. Both have the same Bell curve. What is the difference between the Bell curve and the Gaussian distribution?

I'm looking for the expected value of an exponential Gaussian Y=\text{exp}(jX) \text{ where } X\text{~}N(\mu,\sigma^2) From wolframalpha, http://www.wolframalpha.com/input/?i=expected+value+of+exp%28j*x%29+where+x+is+gaussian E[Y]=\text{exp}(j^2\sigma^2/2+j\mu) If I were to use the...

If I have a variable X whose gaussian distribution is known and let f be a known function, is there a way to compute f(X) (i.e) the resulting gaussian distribution from this? Is the result actually a gaussian distribution?

Homework Statement The problem is given in the attached image. I'm currently trying to work out question one.Homework Equations \phi (k) = \dfrac{1}{2 \pi} \int_{- \infty}^{ \infty} \Psi (x,0) e^{-ikx} dxThe Attempt at a Solution Okay, so the first thing I did was to normalise it, but then I...

Homework Statement We define I_{n} = \int_{-∞}^{∞}x^{2n}e^{-bx^{2}}dx, where n is a positive integer. Use integration by parts to derive:I_{n}=\frac{2n-1}{2b}I_{n-1} Homework Equations Parts formula. The Attempt at a Solution So I'm just stuck here, I'm baffled and confused. Firstly if I...

Dear all, I don't understand why the Cosmic Microwave Background's angular distribution is considered to to a Gaussian random field initially. The rest of the analysis is roughly clear to me, COBE/WMAP/PLANCK measure the CMB Photons and show the temperature fluctuations w.r.t. the mean...

I am trying to calculate the functional for real scalar field: W[J] = \int \mathcal{D} \phi \: exp \left[{ \int \frac{d^4 p}{(2 \pi)^4}[ \frac{1}{2} \tilde{\phi}(-p) i (p^2 - m^2 +i \epsilon) \tilde{\phi}(p)} +\tilde{J}(-p) \tilde{\phi}(p)] \right] Using this gaussian formula...

Homework Statement I'm reading Hinch's perturbation theory book, and there's a statement in the derivation: ...\int_z^{\infty}\dfrac{d e^{-t^2}}{t^9}<\dfrac{1}{z^9}\int_z^{\infty}d e^{-t^2}... Why is that true?Homework Equations The Attempt at a Solution Homework Statement Homework Equations...

Hi All, I'm trying to implement the Gaussian Mixture Model for background subtraction as described by Chris Stauffer and W.E.L Grimson in their paper "Adaptive background mixture models for real-time tracking." I'm having a little trouble with the logic in the step that updates the mean...

Homework Statement Suppose a Gaussian wave packet ψ(x,0) is built out of plane waves according to the amplitude distribution function A_{k} = \frac{Ca}{\sqrt{\pi}}e^{(-a^2(k-k_{0} )^2)} Calculate ψ(x,t) for this packet and describe its evolution. Homework Equations ψ(x,t) =...

Homework Statement "You are now going to show that, in the Gaussian distribution P(x)=Ae^(-Bx^2) the constant A is equal to sqrt(B/Pi). Do this by insisting that the sum over probabilities must equal unity, Integral(P(x)dx)=1. To make this difficult integral easier, frst square it then combine...

What is the expected value of the following expression exp(|z+\mu|), where \mu is a real constant and z=x+jy such that x and y are independent Gaussian random variables each with zero mean and \sigma^2 variance. When I try to take the expectation, I couldn't obtain a gaussian integral, so I...

Is there a closed form expression for the following definite integral? \int_{-∞}^{∞} exp(\frac{-|z|^2}{2{\sigma}^2}-\alpha |\mu + z|)dz where z is complex, and \alpha, \sigma, \mu are real constants. I couldn't obtain an expression similar to Gaussian integral, so I couldn't take the...

Hello all, I have the following equation \mathbf{v}(t)=\mathbf{P}(t)\mathbf{d}+\mathbf{w}(t) where v(t) is a 2-by-1 vector, P(t) is 2-by-2N matrix, d is a 2N-by-1 vector, and w(t) is an 2-by-1 Gaussian process vector where each element is of zero mean and variance N0. What is the probability...

Homework Statement I have an algorithm that implements Gaussian elimination. According to the text, with some modification of the indices and their in the loops, I should be able to have this algorithm perform Gauss-Jordan elimination. I also have to reduce the matrix to reduced row-echelon...

Radioactive Decay Probability? Say you are counting the number of decays and the time of observation is varied. I know that as time increases, the Gaussian Distribution becomes a closer fit to the observed probability than when the time interval takes smaller values because the mean count...

The field makes an angle θ with side 1 and the area of each face is A. In symbolic form, find the electric flux through (a) face 1, (b) face 2, (c) face 3, (d) face 4 and (e) top and bottom. My professor got: a=EAcosθ b=-EAsinθ c=-EAcosθ d=EAsinθ e= 0 I understant why e=0 but for the...

Homework Statement The problem states that you have a spherical shell with inner radius Ri=1 cm and outer radius R0=2 cm. The shell also has uniform charge density of ρ=10-3 N/m3. I found the first few answers of the question already. First was to get the charge of the shell, which is simply...

Homework Statement How can I find the volume of a three dimensional gaussian exp\left [ \frac{-x^2}{\sigma_{x}} \frac{-y^2}{\sigma_{x}}\frac{-z^2}{\sigma_{z}} \right ] ? Since it is a gaussian, the volume should actually extend to infinity. It seems like there should be a simple double or...

Homework Statement This is just a small part of a larger question and is quite simple really. It's just that I want to confirm my understanding before moving on. What are some of the elements of Z[i]/I where I is an ideal generated by a non-zero non-unit integer. For the sake of argument...

At our QM intro our professor said that we derive uncertainty principle using the integral of plane waves ##\psi = \psi_0(k) e^{i(kx - \omega t)}## over wave numbers ##k##. We do it at ##t=0## hence ##\psi = \psi_0(k) e^{ikx}## \psi = \int\limits_{-\infty}^{+\infty} \psi_0\!(k) \cdot...

If I had an integral \int_{-1}^{1}e^{x}dx Then performing the substitution x=\frac{1}{t} would give me \int_{-1}^{1}-e^\frac{1}{t}t^{-2}dt Which can't be right because the number in the integral is always negative. Is this substitution not correct? Sorry if I am being very thick but I...

Homework Statement An insulating sphere of radius R has positive charge uniformly distributed throughout its volume. The volume charge density (i.e., the charge per volume) is ρ. What is the flux through a Gaussian spherical shell of radius R/2 that is totally contained inside the charged...

Homework Statement a cylindrical solid of charge q, radius R, and length H. The Gaussian surface S is a cylindrical shell of radius r and length h, with r < R. Determine the net electric flux given that q = -48Q, R = 4L, H = 3L, r = 2L, and h = 2L (type the integer value, along with the sign...

I have a vector signal, \underline{x}(t), which is afflicted with Gaussian noise \underline{n}(t). I take a finite number, L, of discrete observations and (based on those observations) want to determine whether: (1) Only Gaussian noise is present, \left[\text{i.e. } \underline{x}(t) =...

Homework Statement Consider a uniformly charged sphere (an insulating sphere of radius R,) and a spherical Gaussian surface with radius R/2 concentric to the sphere. What is the total flux flowing through the Gaussian surface? Homework Equations Vsphere= (4∏R^3)/3 Asphere= 4∏R^2 Gauss' Law...

Homework Statement Suppose the one-dimensional field A = Kx * ax exists in a region. Illustrate the validity of the Gaussian theorem by evaluating its volume and surface integrals inside and on the rectangular parallelepiped bounded by the surfaces: x=1,x=4,y=2,y=-2,z=0 and z=3, for a given...

Homework Statement Consider a Gaussian pulse exp[-(t/Δt)^2/2]exp(i*w*t), where Δt is its approximate pulse width in time. Use the Fourier transform to find its spectrum. Homework Equations The Fourier transform of a Gaussian is a Gaussian. If a Gaussian is given by f(t) = exp(-t^2/2)...

I need to work out an expression for the average of a Dirac delta-function \delta(y-y_n) over two normally distributed variables: z_m^{(n)}, v_m^{(n)} So I take the Fourier integral representation of the delta function: \delta(y-y_n)=\int \frac{d\omega}{2\pi} e^{i\omega(y-y_n)} =\int...

Homework Statement Consider the gaussian distribution ρ(x) = Aexp[(-λ^2)(x-a)^2] , where A, a, and λ are positive real constants. (a) Find A such that the gaussian distribution function is normalized to 1. (b) Find <x> (average; expected value) , <x^2>, and σ (standard deviation). (c)...

Homework Statement A point charge of negative polarity is located at the centre of a cubic Gaussian surface with edges of length ##0.5m##. Calculate the electric flux through one of the faces of the surface. What would happen if the charge was moved 10cm to the right? Homework Equations...

Homework Statement 4.10.4 Thin Slab Let some charge be uniformly distributed throughout the volume of a large planar slab of plastic of thickness d. The charge density is ρ. The mid-plane of the slab is the y-z plane. (a) What is the electric field at a distance from the mid-plane when |x| <...

Hello, I was reading about a comparison betw. Gaussian and SI units and the author states "The main advantage of Gaussian units is that they make fundamental physical issues and theoretical relations involving electromagnetic phenomena more clear." Would someone know - is the advantage...

1. Gaussian Please help me calculate some Gaussian integrals in the attached file which are used in my QFT calculations. Thank you very much!

I am doing some calculations in QFT. And, in my calculations, I have to deal with 5 Gaussian integrals as followed. Please help me calculate those 5 integrals. Thank you very much!

Homework Statement Suppose there is a uniformly charged cube with known side length. I then imagine a larger, closed cube surface surrounding it. This larger cube has double the side length and is symmetrical to the smaller cube. Is is practical to use this Gaussian surface to compute the...