What is Gaussian distribution: Definition and 73 Discussions

In probability theory, a normal (or Gaussian or Gauss or Laplace–Gauss) distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is




f
(
x
)
=


1

σ


2
π






e




1
2




(



x

μ

σ


)


2






{\displaystyle f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}}
The parameter



μ


{\displaystyle \mu }
is the mean or expectation of the distribution (and also its median and mode), while the parameter



σ


{\displaystyle \sigma }
is its standard deviation. The variance of the distribution is




σ

2




{\displaystyle \sigma ^{2}}
. A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate.
Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution converges to a normal distribution as the number of samples increases. Therefore, physical quantities that are expected to be the sum of many independent processes, such as measurement errors, often have distributions that are nearly normal.Moreover, Gaussian distributions have some unique properties that are valuable in analytic studies. For instance, any linear combination of a fixed collection of normal deviates is a normal deviate. Many results and methods, such as propagation of uncertainty and least squares parameter fitting, can be derived analytically in explicit form when the relevant variables are normally distributed.
A normal distribution is sometimes informally called a bell curve. However, many other distributions are bell-shaped (such as the Cauchy, Student's t, and logistic distributions).

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  1. C

    I What Is the Probability Atom A Will Emit a Photon Before Atom B?

    For concretness I'll use atoms and photons but this problem is actually just about probabilities. There's an atom A whose probability to emit a photon between times t and t+dt is given by a gaussian distribution probability P_A centered around time T_A with variance V_A. There's a similar atom...
  2. M

    I Grating Resolving Power of Laser Beams with Gaussian Distribution

    All resources I’ve found for grating resolving power assume uniform distribution on the grating and produce airy disks. Resolvance is determined by the Rayleigh criterion where the peak of one wavelength is at the minima of the adjacent one. This definition doesn’t seem applicable for Gaussian...
  3. A

    Probability involving Gaussian random sequences

    How do I approach the following problem while only knowing the PSD of a Gaussian random sequence (i.e. I don't know the exact distribution of $V_k$)? Or am I missing something obvious? Problem statement: Thoughts: I know with the PSD given, the autocorrelation function are delta functions due...
  4. J

    B Sigma Multiplied Gaussian Distribution

    Hi! Say i have two variables that have independent gaussian distributions of probability of being a certain value when i sample them, what is the likely hood that both will land on a 3 sigma value simultaneously? Is there an equation that easily determines that? Also for other combinations like...
  5. P

    Understanding the meaning of "expected fraction" (Statistics)

    The first part of the question asked me to calculate the mean and standard deviation for the number of remain votes in the simple binomial model consisting of total sample size of 2091 people. I believe this is fairly straightforward, it was simply ##E(X) = \mu = 2091(0.5) = 1045.5## votes and...
  6. 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...
  7. B

    I A few questions about doing a Gaussian Fit

    A few questions about doing a Gaussian Fit : 1) Is gaussian fit and gaussian regression the same thing ? 2) I have a gaussian function that will return a list of gaussian numbers giving an initial list length. So if you input 5 you will get: 1,2,6,4,1. My question is if I have an image and I...
  8. tworitdash

    I Can a Gaussian distribution be represented as a sum of Dirac Deltas?

    We know that Dirac Delta is not a function. However, I just talk about the numerical version of it that we use every day. We can simply represent the Dirac delta function as a limiting case of Gaussian distribution when the width of the distribution ##\sigma->0##. $$ \delta(x - \mu) =...
  9. Arman777

    Mutual Information from this Gaussian Distribution

    Let us suppose we are given a Gaussian Distribution in the form of $$p(x,y) \propto exp(-\frac{1}{2}x^2 - \frac{1}{2}by^2 - cxy)$$ What are the equations that I need to use to obtain Mutual Information ?
  10. Z

    What Is the Gaussian Distribution of an Ideal Gas?

    My attempt : $$P(n) = \frac{1}{\mathcal{Z}} Exp[(n\mu -E)/\tau]$$, use $$\lambda = e^{\mu/\tau}$$, then the distribution can be written as $$P(n) = \frac{1}{\mathcal{Z}} \lambda^nExp[-E/\tau]$$ Note that the average number of particle can be written as $$<N>= \lambda \partial \lambda ( log...
  11. mertcan

    A What is the Derivation of the Inverse Gaussian Distribution by Schrödinger?

    Could you help me about the derivation of inverse gaussian distribution? During my search I encountered that it was derived by schrödinger as a result of differential equation solution but I can not find his derivation on internet...
  12. Peter Alexander

    Probability from the tolerance of a capacitor (Gaussian distribution)

    Given the upper data, if the nominal value for capacitance is 33nF and tolerance of 20%, then values can range between 26.4nF and 39.6nF. With the bottom margin being set at 30nF, this means that the interval takes approximately 72% of all values. Is this the correct procedure to solve this...
  13. K

    Deconvolution of fluorescence spectra

    I am trying to make a deconvolution of fluorescence spectra in Matlab. The original spectra is the yellow graph in the figure below. The other two graphs are Cauchy distributions, x and y, that I have manually added to the plot. I would like to write a program that could do this automatically...
  14. Boltzman Oscillation

    What is the integral of this Gaussian distribution?

    Homework Statement Find A in p(x) = Aexp(-λ(x-a)^2) by using the equation 1 = ∫ p(x)dxHomework Equations 1 = ∫p(x)dx The Attempt at a Solution I expand the power of the exponential and then extract the constant exponential to get: Aexp(λa^2) ∫exp(-λx^2)exp(2aλx)dx I don't know how to...
  15. A

    I Sampling from a multivariate Gaussian distribution

    I was watching a lecture on youtube about linear regression and there's a section where it had the statement below (written in purple). Does multiplying by sigma rotate the distribution to make it look like x - N(mew, sigma^2)? Mew in this case is 0 so it doesn't shift the distribution.
  16. D

    I How does the distribution depend on a variable resolution

    Dear all, We were trying to solve the following question but did not quite understand what to do. The question is as follows: The reconstructed invariant mass is usually described by a Gaussian (or Normal) distribution. However, the resolution σ (the width of the distribution) is found to...
  17. A

    I Negative values for Gaussian Distribution

    So in my Physics lab, we divided into groups and our task was to throw darts on a target containing 13 bins. The bins look something like the image below. At the end, our class combined our average, standard deviation, and standard error. I made a Gaussian Distribution and I noticed that the...
  18. senobim

    A Gaussian distribution characteristic function

    Hello, guys. I am trying to solve for characteristic function of normal distribution and I've got to the point where some manipulation has been made with the term in integrands exponent. And a new term of t2σ2/2 has appeared. Could you be so kind and explain that to me, please...
  19. C

    What is the Probability of Hereditary Conditions in a Gaussian Distribution?

    Homework Statement Please help! I'm new to Gaussian and I've been on this problem for hours, I can't crack it at all (no pun intended) can anyone provide a detailed walk through the answers? On average 5% of eggs contain a hereditary condition. Use Gaussian distribution to find the...
  20. E

    A Parameterizing conditional expectations (Gaussian case)

    Consider three jointly normally distributed random variables X,Y and Z. I know that in the Gaussian case E[Z | X,=x Y=y]=xßZX;Y +yßZY;X where ßZX;Y notes the regression coefficient of Z on X conditional on Y (and ßZY;Xis analogously defined). Is the following derivation correct? E[Z| X>x...
  21. S

    Error in summation of spectral components

    HI everyone, Imagine we are sampling of a gaussian signal along time and need to know the power/variance associated with the first N spectral components. So we take our favorite fft algorithm to get the PSD. The error associated with a given estimated spectral component f(w) (w is the...
  22. J

    Integrating Gaussian Distribution (QM)

    Homework Statement I am struggling with one of the end of chapter questions in my QM textbook (see attachment as I don't know how to show calculus on PF). It has thrown me because the chapter introduces some of the key principles in QM by talking about probability but then it randomly chucks in...
  23. A

    Determine the intensity profile and fwhm for gaussian beam.

    Homework Statement A beam of wavelength 600 nm has initially an intensity profile of Gaussian shape with a fwhm of 1 mm. Determine the intensity profile and fwhm 10 meter away.Homework Equations FWHM = λ/(2NA√(1 + I/Is)) The Attempt at a Solution [/B] FWHM1 = 1mm = 600 nm/(2NA√(1 + I/Is))...
  24. Destroxia

    Consider the Gaussian Distribution....?

    Homework Statement Consider the Gaussian Distribution ## p(x) = Ae^{-\lambda(x-a)^{2}} ##, where ## A ##, ##a##, and ##\lambda## are constants. (Look up any integrals you need.) (a) Determine ##A## (I only need help with this (a)) Homework Equations ##\int_{-\infty}^{\infty} p(x)dx = 1##...
  25. Aner

    Fortran [Fortran90]Problem with Gaussian distribution

    Hi, I have a problem in my program and I cannot figure it out. In the last post I had a problem about some arrays, I perfectly resolved it thanks to you, but now my problem is a little bit subtle. I have a subroutine(here I'll post it has a program )that generates random numbers in order to...
  26. QuantumCurt

    Fitting data to a Gaussian Distribution in Excel

    Hey everyone, I'm hoping I can get some input on this. As some of you may recall, I'm currently in an internship at Fermilab, and I've hit a snag in plotting my data. I'm measuring the intensity of the NuMI neutrino beam at multiple different points to look for discrepancies in the measured...
  27. AlanKirby

    Higgs mass - Why is it a Gaussian distribution?

    Hi, so my question is along the lines of the following: If I'm looking at the decay channel H=>ZZ=>4 Muons, why would the resultant distribution for the mass of the parent Higgs be a Gaussian? Is it a case of the peak is the actual value of the Higgs at rest; those of greater mass are Higgs...
  28. J

    Normal distribution starting with a uniformed distribution

    Hi comunity! I need to make a code o a normal distribution of velocities, starting whit a random secuence uniformly distributed between [0,1]. I am using FNT95, with Plato. I want to obtain a ''for'' bucle with I=1,N for the velocities. It is importan for the distribution to have sigma defined...
  29. F

    Convergence of Integral with Real and Imaginary Parameters

    The integral given below is to be computed as a function of real variables x and s. Even a partial answer only for s>0 is very useful. Here is the integral: $$\int_{0}^{\infty}{dk \frac{k^2 e^{-k^2 x^2}}{(k^2 + s)^{3/2}}}$$ Thank you for your help.
  30. 2

    How to Normalize the Quantum Harmonic Oscillator Wave Function?

    when considering the quantum harmonic oscillator, you get that the wave function takes the form psi=ae^{-\frac{m\omega}{2\hbar}x^2} I have been trying to integrate \psi ^2 to find the constant a so that the wave function is normalised, and I know the trick with converting to polar coordinates...
  31. T

    Integration seems gaussian but the answer does not match

    Homework Statement -h^2/2m (sqrt(2b/pi)) e^(-bx^2) d^2/dx^2 (e^(-bx^2)) dx from - to + infinity Homework Equations I tried differentiating e^(-bx^2) twice and it came up weird , I positioned the values and finally cam up with (-2b sqrt(pi/2b)...is there any other way to do it ? The...
  32. G

    Skewed Generalized Gaussian Distribution

    I am looking for more information (e.g., reference, the CDF and descriptive stats) about a four-parameter skewed generalized Gaussian (SGG) distribution. I have come across the PDF for this distribution, but with no reference and not a lot of other information. Here is a snippet... On...
  33. C

    Calculate the standard deviation of Gaussian distribution ,thanks

    Given a one-dimensional Gaussian distribution, distributed as following: f (x) = exp (-x ^ 2 / (2q)) / q / √ (2pi) proof which q is the standard deviation Thanks !The standard deviation is defined by: http://www.mathsisfun.com/data/standard-deviation-formulas.html
  34. L

    Gaussian Distrib: What is Standard Deviation of Mean?

    In my course textbook it is written that "approximately 68% of the measurements from a normally distributed set lie within +-1 standard deviation of the mean value". What do they mean by standard deviation of the mean value? They give a definition for "the mean"(of a set of measurements(data))...
  35. cqfd

    What is the RMS deviation from the true mean for a Gaussian distribution?

    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...
  36. C

    Gaussian distribution other than standard form

    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 ?
  37. C

    Problems with Gaussian distribution

    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...
  38. X

    What's the difference between Bell curve and Gaussian distribution

    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?
  39. S

    Exponential of 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...
  40. S

    Changing the Gaussian Distribution from cartesian to polar coordinates

    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...
  41. E

    Continous Time Gaussian Distribution

    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...
  42. Mandelbroth

    Derivation of the Antiderivative of the Gaussian Distribution

    I'm in a high school pre-calculus class and a statistics class. For the latter, we are given z-tables for some of our tests. I don't like these z-tables. Thus, I decided that a more direct approach (fundamental theorem of calculus) would be more accurate and, more importantly, more fun. My...
  43. A

    Finding the Normalization Constant of a Gaussian Distribution (Griffiths 1.6)

    Homework Statement Consider the Gaussian Distribution ρ(x) = A e^{-λ(x-a)^{2}} where A, a, and λ are constants. Determine the normalization constant A. Homework Equations \int^{∞}_{-∞}ρ(x) dx = 1 The Attempt at a Solution The problem recommends you look up all necessary integrals, so I...
  44. J

    Intuitive explanations for Gaussian distribution function and mahalanobis distance

    Hello I was wondering If anyone could give intuitive explanations for the multivariate Gaussian distribution function and mahalanobis distance? My professor didn't explain these in probability class, they were only defined... Where did the formula come from? Why is the Gaussian function the...
  45. P

    Solving integral - gaussian distribution of cos

    Homework Statement I have to prove: ∫(-infinity:infinity) cos(pi*v/2L)*e^-((L-L_av)^2/sqrt(2pi)*sigma^2) dL proportional to cos(pi*v/2L_av)*e^-(t/tau)^2 tau is some constant, and sigma << L_av. The Attempt at a Solution i can change the integral to 0:infinity, since sigma <<...
  46. M

    Electron Cloud described by a Gaussian distribution

    Homework Statement A cloud of electrons are drifting from a negative plate to a positive plate after being liberated by a laser pulse, (separated by a distance z = 10cm with an original potential difference of 15V) at an instant in time the centre of the cloud has traveled 25mm from the...
  47. S

    Brownian motion: Gaussian distribution

    Homework Statement A grain of pollen shows Brownian motion in a solvent, such that the position x(t) on the x-axis varies with time. The displacement during one second, x(t + 1) - x(t), is measured many times and found to have a Gaussian distribution with an average of 0 and standard devation...
  48. J

    Characteristic Function of Joint Gaussian Distribution

    This is inspired by Kardar's Statistical Physics of Particles, page 45, and uses similar notation. Homework Statement Find the characteristic function, \widetilde{p}(\overrightarrow{k}) for the joint gaussian distribution: p(\overrightarrow{x})=\frac{1}{\sqrt{(2\pi)^{N}det...
  49. A

    Bounded/Truncated Gaussian distribution

    Dear all, I have a problem in understanding how to bound a Gaussian distribution. LEt me describe the problem at hand: Let's say that we have a Gaussian distribution in the x-coordinate and a Gaussian distribution in the y-coordinate. Further, assume that the independent random variables x...
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