Questions tagged [gaussian-integral]

For questions regarding the theory and evaluation of the Gaussian integral, also known as the Euler–Poisson integral is the integral of the Gaussian function $~e^{−x^2}~$ over the entire real line. . It is named after the German mathematician Carl Friedrich Gauss.

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gaussian integral without the error function

here is my attempt.is it correct? $$I = \int e^{-x^2}dx$$ let $u = e^{-x^2}$ and $v=x$ then : $$I = xe^{-x^2} - \int -2x^2e^{-x^2} dx ; J = \int -x^2e^{-x^2}dx$$ here is where I'm in doubt am I alowed ...
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Why the dummy variable $y$ in the calculation of the gaussian integral as follows?

I don't understand why you have to use a different variable when squared the first integral? It is commonly glossed over to explain this.
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Integrating bi-variate Gaussian density with respect to their correlation coefficient

I'm wondering if there is any literature that deals with the integral of the following type $$ \int \dfrac{1}{2\pi\sqrt{1-\rho^2}} \cdot \exp(-\dfrac{(a^2+b^2+2\rho ab)}{2(1-\rho^2)}) \cdot d\rho $$ ...
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Gaussian integral in 3rd dimensions

I have been wondering about computing $(1/3)!$ and using the Gamma function. After substituting for $x=t^{\frac{1}{3}}$, I got $\int_{0}^{\infty}e^{-x^{3}}dx$. May I know if there is any way to ...
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How to prove or calculate $E[\int_{t_{i-1}}^{t_i} e^{-μ({t_i}-s)}\sigma B_s |x_{t_{i-1}}]=0$?

$B_s$ is brownian motion. Because $\int_{t_{i-1}}^{t_i} e^{-μ({t_i} -s)}\sigma dB_s $ has a Brownian component, it is normally distributed with the mean zero according to Taylor & Karin 1998 's ...
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Evaluating the Gaussian-like integrals $\int dx\, x^{-n} \exp(-(x-b)^2)$

Is there a known form for indefinite Gaussian integrals of the form $$\int dx\, x^{-n}\exp(-(x-b)^2) $$ where $n$ is a positive integer and $b$ is some constant? Mathematica cannot solve integrals of ...
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Very fast but inaccurate estimations of multivariate Gaussian integral over a hypercube

$\def\Z{\mathbb{Z}}\def\R{\mathbb{R}}\def\A{\mathcal{A}}\def\N{\mathcal{N}}$ I'm working on 4D positive real values, i.e. $\R^4_{\geq 0}$, where it is gridized with hypercubes of side length $a > 0$...
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Simplify the multiplication of Two Gaussian Function

I am reading this article https://www.mathematica-journal.com/2014/12/08/evaluation-of-gaussian-molecular-integrals-4/. The derivation at equation (9) confused me. At equation (8), it's $$ NE=E_{AB}\...
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Application of Gaussian Integration Technique

In order to solve the Gaussian integral $\int\limits_{-\infty}^{\infty} e^{-x^2}\,dx$, we set this value to $I$, square, and evaluate the following double integral in polar form ($I=\sqrt{\pi}$). Can ...
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Jacobian in multidimensional Gaussian integral

For the first integral shown in: reference for multidimensional gaussian integral It is mentioned that the Jacobian is 1, how is this the case? By following the calculation I found it to be $det(diag(...
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Fourier transform of a gaussian of width sigma

Introduction: let's define the Fourier transforma F of a given function f as: $ F(k) = \frac{1}{\sqrt{2.\pi}}\int\limits_{-\infty}^\infty f(x)exp(ikx)dx $ I'm currently sovling a problem for which the ...
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Reference Request: Integral of Gaussian over Unit Sphere

I am looking for a reference for integrals of the form \begin{equation} \tag{1} \int_{S^{n-1}} \mathcal{N}_{\omega} ( \mu , \Sigma ) d \omega \end{equation} where $S^{n-1}$ is the sphere in $\mathbb{R}...
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How is it possible to find this estimate of the heat semigroup?

In this paper, in the proof of Lemma 3.3, there is the passage $$S(t-s)|x|^{\sigma/1-k} \geq C_1(t-s)^{\sigma/2(1-k)}.$$ Here $S(t)$ denotes the heat kernel. I tried to do the calculations to check ...
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Integration of Gaussian with nested integration domains

I am trying to integrate the following $$ I_j=\frac{1}{(2\pi)^{n/2}}\int_{-\infty}^{+\infty} \mathrm{d}x_1 \int_{-\infty}^{x_1} \mathrm{d}x_2 \int_{-\infty}^{x_2} \mathrm{d}x_3 \dots \int_{-\infty}^{...
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Find flux of the vector field $\vec A$ through the surface of a sphere with radius R and center on the origin

The vector field is given by the spherical coordinates $\vec A = kr^3\,\vec {e}_r$, where $k$ is a constant and $r=\sqrt {x^2+y^2+z^2}$. I thought about using the Gauss-Ostrogradski theorem, where the ...
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Gaussian Integral like $\exp{[-r^2 - r_\alpha^2 +2 r r_\alpha \cos{(\theta-\theta_\alpha)}]}[r -r_\alpha \cos{(\theta-\theta_\alpha)}]f(r,\theta)$

I have a Gaussian kernel that I wish to evaluate $$\int_{0}^{\infty} \int_{0}^{2\pi} \exp{[-r^2 - r_\alpha^2 +2 r r_\alpha \cos{(\theta-\theta_\alpha)}]} [r -r_\alpha \cos{(\theta-\theta_\alpha)}]f(r,\...
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An expectation based on the mean field rule

I am reading a paper about mean field rule and message passing, there is a equation in this paper: $\exp\{E_{b(z_j)}[\log\mathcal{N}(z_j;r_j,\epsilon^{-1})]\}$ $\propto\epsilon\exp\{-\epsilon(\left|...
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Computation of the Laplace transform of the Gaussian heat kernel

Currently I'm interested in the Laplace transform of the Gaussian heat kernel $$ k_t(x):=\frac{1}{(4\pi t)^\frac{d}{2}}e ^{-\frac{|x|^2}{4t}}. $$ Using the Laplace transform $$ G_\lambda(x)=\int_0^\...
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Gaussian bigger than other gaussians

Let $N_1, N_2, N_3$ be three independent Gaussian random variables. Is there a closed form for $$ \mathbf P \big[\{N_1 \ge N_2\} \cap \{ N_1 \ge N_3\}\big]=\int_{\mathbb R} \Phi_2(y)\Phi_3(y) p_1(y)\,...
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Simplify $\Pr(Z\ge P(X))$ with $Z\sim N(0,1)$

Consider a discrete rv $X$ with probability function $p(x)$ and an independent $Z\sim N(0,1)$ I am trying to write the probability \begin{align} \Pr(Z\ge p(X)) \end{align} as a single Q-function $Q(x)\...
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Integrating Probability Density Function for Independent Gaussian Variables with Non-Zero Means

I am trying to solve the following problem and have not been able to. Any help would be much appreciated. The problem is explained below. $$P(\theta_{B})=\int_0^{\Theta_{B}}\frac{\theta_{\gamma}}{\...
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Coulomb Potential and laplace operator

How to proof the equality $\int_{S_{a}^{n-1}}\nabla u.d\sigma= c *vol (S^{n-1})$. Hi all, I am reading a text in portugues about PDE, is about Laplace operator and Coloumb Potential my specifics ...
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How do I convert the integral $\int_0^\infty \int_0^\infty (x^2 + y^2)e^{-(x^2+y^2)} \ dxdy $ into polar coordinates?

Solve the following integral $$\int_0^\infty \int_0^\infty (x^2 + y^2)e^{-(x^2+y^2)} \ dxdy $$ For me, it is clear that we can use polar coordinates to solve this integral quickly (and yes, we can do ...
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Gaussian integral using Euler/Jacobi theta function and $r_2(k)$ (number of representations as sum of 2 squares)

The Euler/Jacobi theta function (using the notation of this question) is $\vartheta_3(\tau) := \sum_{n\in \mathbb Z} q^{n^2}$ where $q = e^{2\pi i\tau}$ is the nome. The square $(\vartheta_3(\tau))^2$ ...
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Does the expectation of any bounded function on Gaussian variables exist?

I want to prove the expectation of the function $f(X)$ exists, where $X$ is a Gaussian variable. Given that $f(X)$ is bounded between $[-c, c]$, with $c$ being some positive constant, does the ...
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Single integral over bivariate normal distribution

I have $(X, Y)$ are bivariate normal with $\mu_X=log(160), \mu_Y =log(165),\sigma_x=0.05=\sigma_y$, and $\rho=0.5$. After a change of variables, I want the marginal density $f_U(u) = \int_{0}^\infty\...
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An complicated integral involving Erfi function

Sir, While, studying the diffraction of the Gaussian beam through apertures, I have faced the following integral (as an expression of diffracted field), $$I= \int_0^R \exp(-\alpha r^2+i\beta r)\bigg[\...
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Efficiently approximating the integral of $\operatorname{erf}(x y) \exp(-x^2)$

I need to efficiently approximate this integral which represents the Gaussian-weighted area of a right triangle whose three points are the origin plus two points that form a vertical "edge", ...
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Grassmannian gaussian integral

$A$ a 2 by 2 matrix and let it be anti symmetric. Then $detA=A_{12}^2$. The integral is said to hold: $$\int d\hat\theta\int d\theta e^{\theta ^T \cdot A \cdot \hat\theta}=detA$$ Where $\hat\theta ,\...
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Divergence Theorem for 1D

As all of us know, in divergence theorem we have This: this means instead of integrating a divergence of a vector over a volume, you can integrate that vector over the surface. But I'm wondering what ...
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3 votes
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Interesting $\arctan$ integral

In a generalization of this problem: Integral involving product of arctangent and Gaussian, I am trying to calculate the integral $$ I(a,b) = \int_{\mathbb{R}^2} \arctan^2 {\left( \frac{y+b}{x+a} \...
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How to compute the following integral with the gauss formula?

I have the following question. I need to compute the following integral $$\int_A xy+yz+zx \,\,\,dxdydz$$ for $A=\{(x,y,z)\in \mathbb{R}^3: x,y,z\geq 0, x^2+y^2+z^2\leq 1\}$ with the gauss formula. ...
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Integrating a two-dimensional Gaussian function with definite limits

I struggle with following problem. A two-dimensional Gaussian function $f(x,y) = \frac{1}{2 \pi s^2} e^{-(x^2+y^2)/(2s^2)}$ is given and shall be integrated within the limits of $a=-\sqrt{2\mathrm{ln}(...
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Computing the Fourier Transform of $\exp(-\log(x)^2)$

I would like to use the approach of R. E. Fredericksen and R. F. Hess, “Estimating multiple temporal mechanisms in human vision,” Vision Res., vol. 38, no. 7, pp. 1023–1040, Apr. 1998, doi:10.1016/...
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How do I solve this quadruple integral that involves gaussian like terms?

I'm trying to solve the following integral: $$ \int_{x} \int_{y} \int_{z} \int_{w} \exp \left(-\frac{1}{2}\left\|\left(\begin{array}{l} y-x \\ z-y \\ y-z \end{array}\right)\right\|^{2}\right) d x d y ...
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Find integration limit given the value of the (2D) integral

(Previously asked on Stackoverflow - now deleted - it was pointed out that it makes more sense to ask my question here.) I am trying to solve for R_j(I_0) given in Eq. 12 of this paper (see also Eq. ...
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What is the second step of this standard normal MGF derivation?

On this page, it shows the derivation for the MGF of a standard normal density. I don't get why the second step has a $\sqrt{2}\sigma$ which is pulled out from the integral. Can anyone help me ...
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Numerical integration of $\exp(-x^2)$ in a bounded region

Is there a possible way to integrate $\exp(-x^2)$ in a bounded 2D triangular region numerically with minimal number of Gauss points? The Gauss-Hermite quadrature scheme is suitable for unbounded ...
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About gaussian distribution - calculation of variance (change of variables step)

According to these slides, in slide 3: Why we do not change the x of exp with (x-mu)^2 ? I did not understand this part.
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Solving Gaussian Integral

I learned that a Gaussian Integral is \begin{equation} \int_{-\infty}^{\infty}xe^{-2ax^2}dx=0 \end{equation} Because of the odd function symmetry. But if I shift the $x$ to \begin{equation} \int_{-\...
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Limit of integral with Parseval

I am trying to evaluate the limit $$ \lim_{n \to \infty} \int_{-\infty}^{\infty} \frac{\sin(nt)}{\pi t}f(t) \,dt, $$ where $$ f(t)=e^{-t^2+2t}. $$ Since I am working with Fourier transforms I thought ...
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Is the following cartoon mathematically correct?

I saw the following cartoon about probability and statistics the other day: The joke in the above picture being that "the top 1%" (i.e. the integral of the gaussian probability distribution ...
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How do you integrate the volume of 3 two-dimensional gaussian distributions with 4 hyperbole boundaries rotated 90 degrees at various intervals?

Suppose you have 3 two-dimensional gaussian distributions added up on each other pushed to be within 4 identical hyperbolas rotated in relation to each other by 90 degrees and their most proximal ...
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When evaluating a definite integral by integrating by parts is the following allowed?

$$= \int_a^b f(x)g(x) \, dx = \int_a^b f(x)\, dx\cdot [g(x)]_a^b\, -\int_a^b \bigg[ g'(x) (\int f(x)dx\bigg] \, dx $$ $$ = C - \int_a^b \Bigg[g'(x)\cdot\int_a^b f(x)dx\, \Bigg]dx \, $$ where C is ...
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Derive the quadrature rule of the form $\int_{-1}^1 f(x) dx \approx w_0 f(\alpha) + w_1 f'(\beta)$

From a practice problem Problem Statement Let the interval $[a,b]$ be divided into $n$ subintervals by $n+1$ points $x_0 = a , x_1 = a+h$, $x_2 = a+2h , \dots ,x_n = a+nh =b$. Derive the quadrature ...
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Relationship between $\gamma_n(A + B)$, $\gamma_k(A)$, and $\gamma_{n-k}(B)$, where $A \subseteq V$, $B \subseteq V^\perp$, and $\gamma_n := N(0,I_n)$

Let $\gamma_n = \gamma_1^{\otimes n}$ be the standard gaussian measure on $\mathbb R^n$ and let $V$ be a $k$-dimensional subspace of $\mathbb R^n$ with orthogonal completment $V^\perp$. Let $A$ and $B$...
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Integrating the gaussian kernel on unbounded curves in $\Bbb C$

It is well known (e.g. using Fourier transform) that if $\beta\in\Bbb C$ with $\Re\beta>0$, then $$ \int_{\Bbb R}e^{-bu^2}\,du=\sqrt{\frac{\pi}{\beta}}\;. $$ Once chosen a branch for the square ...
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Given $|\int f(\zeta)\,d\zeta-1|<\epsilon$, what can we say about $\int|f(\zeta)|\,d|\zeta|$?

Consider the following unbounded curve in $\Bbb C$ $$ \Gamma=\{x+ih(x),\;\;x\in\Bbb R\} $$ where $h\colon\Bbb R\to\Bbb R$ is a bounded $\mathscr C^1$ real function, such that $h'(x)\to0$ as $|x|\to\...
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How do I compute this integral on the set A?

I have the following problem: Compute the integral $\int_{\partial A} \langle F,\nu \rangle dS$ where $\nu$ is the normal vector and $$F(x,y,z)=(2x-3y+z,\,\,\,x-y-z,\,\,\,-x+y+2z)$$ and $$A=\{(x,y,z):...
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Partial integration of modified form of Ampere's law

After thinking and searching of/for an answer for an eternity, you are my last hope: The given equation: $$ rot(µ^{-1}rot(\vec A))-\epsilon\omega^2*\vec A=\vec J $$ should be multiplied with a ...
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