Gaussian Definition and 706 Threads

Homework Statement Solve the following system of equations using Gaussian Elimination: -2a+5c=1 a+2b-c=2 3a-2b=3 Homework Equations N/A The Attempt at a Solution Augmented matrix (-2 0 5 | 1) R2 → R2 +R1/2 (1 2 -1 | 2) R3 → R3 - 3R2 (3 -2 0 | 3) (-2 0 5 | 1) Swap R2 and...

Hi, I need help with the gaussian function in 3D. I'm using the form: z=A*exp( (x-xo)^2/(2σx^2)+(y-yo)^2/(2σy^2)) I know that A is the amplitude and xo,yo are the center coordinate. I found that formula on http://en.wikipedia.org/wiki/Gaussian_function and they say that σx and σy are...

Homework Statement Homework Equations The Attempt at a Solution The issue I'm having here is that the problem should be able to be done rather quickly. I can see how to solve for <H> using the operator, but there's a quick way that I'm not picking up on. I thought about solving <H> = <p^2> /...

I am new to lasers. In the expression for q parameter, 1/q = 1/R - j(λ/πw^2) how did we come to know that w(z) is a measure of decrease in field amplitude E with distance? I can't feel it. Does the gaussian beam itself contract to the minimum diameter? Shouldn't a lens be required for this...

Homework Statement A= <1, -3, -1, 1> <2, -5, 0, 1> <-3, 5, -6, 3> What I did was Row2 = 2*Row1 - Row2 which renders Row2 as: <0, -1, -2, 1> However in the solutions, Row2 was given as: <0, 1, 2, -1>, which appears to be R2 = -2R1 + R2 I'm guessing it makes no real difference...

well, i have to prove that the inv. Fourier transform of a gaussian (e^(-(k^2/2)) is a gaussian, i know some elementary complex analysis(never actually taken a class in it), not well enough, it seems, to find the solution to this. I tried to integrate over a circular contour, and let the radius...

I can not find any useful online tool that solves you a system of equations using Gaussian elimination and LU decomposition. So just a system like: -3X+4X+9X+4x=-2 9X+2X+1X-5X=2 etc. Just about 4 lines. several lines. So basically an online tool where you can just plug in the...

What's the kinetic energy uncertainty for gaussian wave packet ψ(x)=((α/pi)^(1/4))exp(-αx^2/2)?

Homework Statement Calculate <x> for the Gaussian wave packet \psi(x)=Ne-(x-x0)/2k2 Homework Equations \left\langle x \right\rangle = \int dx x|\psi(x)|2 The Attempt at a Solution So I've been reviewing for the up-coming midterm and I've had the painful realization that I'm...

Homework Statement Determine the value of k for which the system x + y + 2z = 3 x + 2y -4z = -3 2 + 8 + kz = -29 has no solutions. Homework Equations None The Attempt at a Solution Converting to a matrix 1 1 2 3 1 2 -4 -3 2 8 k -29 r2=r2-r1 1 1 2 3 0 1 -6 0 2 8 k -29...

http://www.calctool.org/CALC/phys/optics/f_NA Does anyone know how they calculate the spot size (Gaussian FWHM) at this website? For a plane wave, the focused spot size it approx=2*wavelength*f/(pi*d0) f=focal length of lens d0=collimated beam diameter before lens but how do you...

Homework Statement Say you have a nonconducting spherical shell. A is the inner radius and B is the outer radius. If you wanted to set up an integral to find the flux, would the gaussian surface include B, or be just inside it? That is, would the limits of integration go from A to B? If...

Is there an intuitive way to see why Gaussian wave packets are the only ones that satisfy \Delta x \Delta p = \frac{\hbar}{2}?

Homework Statement A uniformly charged, straight filament 7.00 m in length has a total positive charge of 2.00 \muC. An uncharged cardboard cylinder 2.00 cm in length and 10.0 cm in radius surrounds the filament at its center, with the filament as the axis of the cylinder. Using reasonable...

Homework Statement I have been given the function \varphi_{G}(z,t) = \frac{2}{\Delta k_0\sqrt{\pi}} \int_{-\infty}^{\infty}dk\ e^{-\frac{4(k-k_0)^2}{\Delta k_0^2}}e^{i(kz-\omega t)} and been told to do the integration and then to specify the phase and group velocity of the wave package. I...

how to integrate this equation? ∫∫e^-x^2 e^-x'^2 / |x-x'| dx^3 dx'^3 lower limit is 0 and upper limit is inf for x and x' the result is an error function. But I would like to know the details of the process of integration. some one suggest me to change the variable to x+x'=u...

Homework Statement start with the wave function \Psi(x,0) = Ae^{-cx^{2}}e^{ikx} where A,c, and k are real constants (and c is positive) i) Normalize \Psi(x,0) ii) Determine \Psi(x,t) and |\Psi(x,t)|^{2} Homework Equations The Attempt at a Solution I normalized it to get...

Help with Gaussian integration problem please :) Homework Statement Compute the improper integral \int^{\infty}_{-\infty}x^{2}e^{-x^2}dx given \int^{\infty}_{-\infty}e^{-x^2}dx=\sqrt{\pi}. Homework Equations Just the rule for doubly-improper integrals I guess...

Hi all, I'm considering a problem of a bunch of particles (say electrons) arriving on a plate. If the plate is orthogonal to the direction of motion of the particles their distribution (on the plate) is a gaussian distribution centered in a point (x,y) of the plate with standard deviation...

Hey, all. I need help computing a particular integral that's come up in my quantum mechanics homework (Griffith 1.3 for anyone who's interested). It involves integration of a gaussian function: \sqrt{\frac{\lambda}{\pi}}\int_{-\infty}^{+\infty} \! {x}{e^{{-\lambda}{(x-a)^2}}} \, dx Where...

I'm working on an exam that Michael Artin once gave, where one of the questions is basically, Consider the homomorphism from Z[x] to Z[i] given by x --> i. What does this homomorphism tell you about the ideals of Z[i]? So far I haven't come up with anything. I know in advance that the...

I have a 3d data cube. For every point I measure the property A which is a gaussian variable of mean m and variance s and it's also a function of the density d at every point. A(x,y,z)=f(d(x,y,z))e^(-(X-m)^2/(2\sigma^2)) X is a random number. Now let's say I want to sample the...

Hello I am having trouble deriving using block matrix notation the conditional pdf of two joint Gaussian vectors: I assume that it just involves some re-arranging of eq 1 (attatched) but am unsure if taking the inverse of the resultant matrix in eq 1 is valid and if the order of...

Hi, My understanding in this is limited, but I'm trying to create a set of Gaussian curve response functions that will be convoluted with some signal data. I've dug all around the internet/textbooks but I'm having trouble finding a way to create the curve with the data I have. All I know is the...

Homework Statement Not really a homework problem, but something that I was curious about. I was thinking about how a calculator finds the area under the standard normal distribution, and I started to assume that it most likely has the antiderivative for the function that makes the standard...

Hi all, I am having trouble with the concept of joint pdf's. For example - a set Z1,Z2,...ZN are each gaussian rv. Let Z1~N(0,1), let X be +1 or -1 each with probability 0.5. Z2=Z1X1, so Z2 is ~N(0,1). (I assume this to be As Z2 is just Z1 multiplied by a simple factor, an instance...

I need to use the four-point Gaussian quadrature rule to do some intense numerical calculations. Could anyone link to this page where it's written out explicitly over an [a,b] interval. I haven't been able to find it, I'm trying to derive it now but it's crucial that I'm 100% correct. I haven't...

Hi! Does anyone know how to solve the following integral analitically? \int^{1}_{0} dx \ e^{B x^{2}} J_{0}(i A \sqrt{1-x^{2}}), where A and B are real numbers. Thanks!

Hi, everybody Let n_1 ~ Poisson ( \lambda_1 ) and n_2 ~ Poisson (\lambda_2). Now define n=n_1-n_2 . We know n has "Skellam distribution" with mean \lambda_1-\lambda_2 and variance \lambda_1+\lambda_2, which is not easy to deal with. I want to find the Pr(n \geq 0) . Is it...

Hello. In a software application I am attempting to smooth a data set by convoluting it with a discrete Gaussian kernel. Based upon information garnered online, I've been using this Mathematica command to generate the kernel: kern = Table[Exp[-k^2/100]/Sqrt[2. Pi], {k, -range, range}]; where...

What is the PDF of the exponential of a Gaussian random variable? i.e. suppose W is a random variable drawn from a Gaussian distribution, then what is the random distribution of exp(W)? Thank you!

1. Consider a uniformly charged conducting plate with an infinite extent, where the sur- face charge density is surf . We introduce a Gaussian surface as a cylindrical pillbox of ra- dius a, half-way submerged into the surface. The figure below shows a chunk of the infinite plate and the...

Hi Experts, I'm working in industry and have an application requiring some expert knowledge on statistics/probability. I have a probability distribution function (PDF) for a Gaussian random variable. I know the standard deviation of the PDF. I also know total number of experiments conducted...

If i have the mgf of X and the mgf of Y where X~N(mx,vx) and Y~N(my,vy) and X and Y are independent , then if i want to show that aX +bY ~ N(amx+bmy , a^2vx+b^2vy) how would i do this - need to be able to do the convolutions way and the mgf's way, for the mgfs way is it just, mgf(ax+by) =...

Hello, my question ; Suppose X1 and X2 are independent random variables, each with the standard gaussian distribution. Compute, using convolutions the density of the distribution X1 + X2 and show that X1+ X2 has the same distribution as X * root2 where X has standard gaussian distribution...

I'm wondering if there was a table of moments for a Gaussian Distribution, I found one up to the fourth moment U \sim N(\mu, \sigma^2) E[U^2]=\mu^2+\sigma^2 E[U^3]=\mu^3+3\mu\sigma^2 E[U^4]=\mu^4+6\mu\sigma^2+3\sigma^4 I'm doing a problem right now and i need the 8th moment.

[FONT="Georgia"][FONT="Tahoma"]Homework Statement Assume that we have a metric like: ds^{2}=f dr^{2}+ g d\theta^{2}+ h d\varphi^{2} where r,\theta , \varphi are spherical coordinates. f,g and h are some functions of r and theta but not phi. Homework Equations How can I calculate...

Can you tell me what spectral weight is? If I have a spectrum that I measure, how can I get spectral weight? For simplicity, assume that the spectrum is a Gaussian curve. Thank you so much!

I have been calibrating a sub-micron particle sizer with 1 um (1000 nm) standard. After testing the standard, the test results on my print-out: (X25 = 812.7 nm, X50 = 977.7 nm, X90 = 1389.0 nm) According to USP, the limits are +/- 6% of the reference standard values for X25, X50, and X90...

Hi, Suppose that an n-dimensional vector \mathbf{z}=\begin{pmatrix}z_1&z_2&\cdots & z_n\end{pmatrix}^T is characterized as a zero-mean circularly symmetric complex Gaussian random vector. What is the distribution (the probability distribution function PDF) of this vector in both: complex and...

Homework Statement Prove that the series \sum_{n=0}^\infty e^{-n^2x^2} converges uniformly on the set \mathbb{R}\backslash\ \big] -\epsilon,\epsilon\big[ where \epsilon>0Homework Equations n/aThe Attempt at a Solution I have tried using Weierstrass M-test but I have not been able to find a...

I just developed this model to describe an ecological process but have trouble solving the equation. My first question is: is the form even analytically solvable? And if so, what steps / references should I resort to? 'a', 'delta' and 'sigma' are all constants. *Current reference book...

what is the advantage of using a gaussian distribution of noise (white/colored) in a system over any other distribution ??

Homework Statement A metal sphere of radius a is surrounded by a metal shell of inner radius b and outer radius R, as shown in the diagram below. The flux through a spherical Gaussian surface located between a and b is 1.20Q/εo and the flux through a spherical Gaussian surface just outside R...

Hi, I was wondering if having some training in algebraic number theory is a must for even starting to work with Gaussian Integers, or one can work with them with some knowledge of abstract algebra, like group, ring and field theory knowledge (i.e. 1 year of undergraduate abstract algebra)...

Homework Statement A long coaxial cable consists of an inner cylindrical conductor with radius a and an outer coaxial cylinder with inner radius b and outer radius c. The outer cylinder is mounted on insulating supports and has no net charge. The inner cylinder has a uniform positive charge per...

The question is: for a gas of weakly interacting molecules show that <v_x>=0 where <v_x> is the average velocity in the x direction. the probability of a molecule having a velocity v is given by: p(v_{X})=\sqrt{\frac{m}{2\pi kT}}e^{-\frac{mv_{x}^{2}}{2kT}} The above is a Gaussian...

In terms of programming. Like, let's say you have F1+F2+F3+15=0 and you move the 15 to the other side, just as you're supposed to, which would be F1+F2+F3= -15. Is that what they mean by pivoting? The code moves the 15? thanks

Hi, I am interested in the product of a Gaussian random variable with itself. If X is Gaussian then what is X^2? We know that the resultant variable of the product of two independent Guassian variables is still Gaussian but I am afraid that this is not true when you multiply it with itself. Is...

I'm trying to show that the electric field for an infinitely large plate is E = \frac{\sigma}{2\epsilon_o} I was wondering why you can only use a cylindrical gaussian surface and not a cube? Thanks