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Questions tagged [gaussian-integral]

For questions regarding the theory and evaluation of the Gaussian integral, also known as the Euler–Poisson integral .

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integration involving Gaussian PDF and CDF, with a scale and offset [duplicate]

Suppose $\phi, \Phi$ are PDF and CDF for a $1$-dimensional normal Gaussian distribution, and $a,b>0$ are arbitrary constants. Is there a way to compute this integral analytically? $$\int_{-\infty}^{...
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Solve the differential equation $\frac{dy}{dx}=5+xy+2x+2y$

Solve the differential equation $$\frac{dy}{dx}=5+xy+2x+2y$$ Given $y(0)=0$ My try: The given equation can be written as: $$\frac{dy}{dx}=1+(x+2)(y+2)$$ Letting $X=x+2$ and $Y=y+2$ we get $dy=dY$ ...
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54 views

When does “Gaussian integrability” imply regular integrability?

Let $\varphi:\mathbb R\to\mathbb C$, and suppose that the limit $$\lim_{\sigma\to\infty}\int_{-\infty}^\infty \varphi\left(x\right)\exp\left\{-\frac12\cdot\left(\frac x\sigma\right)^2\right\}\,dx$$ ...
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30 views

Is there an analytic form for the squared error of the difference of two univariate Gaussians?

Using anchored ensembling it is possible to estimate the mean $\mu$ and the variance $\sigma^2$ of a output. I had the insight than if I then sampled from $N(\mu,\sigma^2)$, I could estimate both the ...
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32 views

Derivation of a closed-form solution for the integral of a 3D Gaussian over the *positive* reals

in a post from over four years ago, Przemo gave the following formula for the integral over a Gaussian function over the positive reals in three dimensions (denoted here as $\mathbb{R}^3_+$) with ...
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Matching widths of two different functions

Say I have a rectangular function as follows $$f(x) = \begin{cases} 1 & \text{$|x|$$\leq$$x_0$} \\[2ex] 0 & \text{otherwise} \end{cases}$$ I want to match its width (which in this case is $...
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1answer
24 views

Extreme points of bounded measurable functions with bounded $\ell_2$ norm

Define $$\mathcal{F}:=\{f:\mathbb{R}^n\rightarrow[-1,1]:~\mbox{$f$ is measurable and }~\mathrm{E}_{x_1,\ldots, x_n\sim N(0,1)}[|f(x_1,\ldots, x_n)|]\leq c\},$$ where $c<1$ and $x_1,\ldots x_n$ ...
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1answer
28 views

how to show: Is $u$ harmonic, so are following identities true

Let $V \subset \mathbb{R}^n , 2 \leq n $ be a set, where you can apply Gauß's Theorem. To show: Is $ u \in C^{(2)}( \bar{V} ) $ harmonic on $V$, then: $$ \int_{ \partial V} \frac{ \partial u }{ \...
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Need help in solving an equation involving volume, single and double layer potentials

Let be $V \subset \mathbb{R}^n $, $ 3\leq n$ an open set, where you can apply Gauß's Theorem. To show is, that for all $ U \in C^{(1)} ( \bar{V} ) \cap C^{(2)} (V) $ with bounded 2nd derivatives ...
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38 views

Asking for a function name [closed]

Is there any name for the function $z=\exp(-(x^2+y^2))$?
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14 views

How do I determine the weights and abscissas in the 1 and 2-point Gauss quadrature given a weight function?

Determine the weights and abscissas in the 1 and 2-point Gauss quadrature formulae for $\int_{0}^1 f(x)w(x)dx$ with weight $w(x) = − \ln x$. I'm pretty confused on how to approach this problem with a ...
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21 views

Problem on computing the Fourier transform of the Gaussian

The problem sounds like this. Show that $s\to\int_\mathbb{R}e^{-(x+is)^2}dx$ is constant wrt $s\in\mathbb{R}.$ Then use this fact to shot that $\mathcal{F}(e^{-a|x|^2})=e^{-\frac{|x|^2}{a}}$ ...
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Solving an integral with four point Gauss-Chebyshev

I am struggling with this question as in our notes it shows how to solve Gauss-Chebyshev integrals of the form $ \int_{-1}^{1}\frac{f(x)}{\sqrt{1-x^2}}dx$ , however this is different. $$\text{Solve ...
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1answer
45 views

Show that $q$ must divide one of the prime integer factors of $N(q)$.

The following is from Aluffi's Algebra Why this is true? : .. since q is prime in Z[i] ⊇ Z, q must divide one of the prime integer factors of N(q).
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1answer
32 views

Bisection Newton - Quadratures

The problem states the following: Find with at least 10 digitis of precisión the roots of the following equation: $\int_x^{x^2} \!e^{-t^2}\,\mathrm{d}t = x^5 -3x^2 + 1 $ in the closed Interval [-1,1]. ...
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Multi-dimensional gaussian integral with non-symmetric & non-hermitian coefficient matrix

There is a commonly used formula in quantum field theory, \begin{equation} \int\prod_{i} \frac{dz_{i}^{\dagger} dz_{i}}{2\pi} e^{-z^{\dagger}Az}=\frac{1}{\det(A)}, (1) \end{equation} where $z=x+\text{...
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2answers
64 views

Verifying Green's theorem for a function

Let $G = \{ (x,y) \in \mathbb{R}^2 : x^2+4y^2 >1, x^2+y^2 < 4 \} $ $ \int_G x^2+y^2 d(x,y) $ I want to verify Green's Theorem : $ \oint_{ \partial G } f n ds = \int_G \operatorname{div}f\, ...
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Developing a cross product of tensors within integrals

I read in a book the following unproven statement: $\int_{s} u\times A n \, ds = \int_v ( u\times \nabla\cdot A + \mathcal{E}: A^T ) \, dv$ with a: 1st order tensor, n: normal vector of s, A: 2nd ...
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1answer
53 views

Simplify double sum

Does the following expression has a closed form \begin{align} E \left[ \| Z\|^k \exp(t \|Z\|^2)\right] \end{align} for $k$ is even and $Z$ is standard normal vector. For the case of $k=2$ the ...
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51 views

Integral of two Bessel functions product times Gaussian

Does anyone have a clue about how to solve this integral? Will it have a closed form? $\int_0^\infty e^{-x^2}J_n(ax)J_n(bx)dx$ I've been searching materials and papers for a while, and did find ...
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Closed form solutions to a Gaussian equation

Let $\phi(t) := \frac{1}{\sqrt{2\pi}}\exp\{-t^2/2\}$ be the standard Gaussian pdf function and $\Phi(t) := \int_{-\infty}^t \phi(u)du$ be the Gaussian CDF function. Consider equation $$ \Phi(x) + \phi(...
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Inner product between Gaussian radial basis functions

Let $\phi : \mathbb{R}^n \rightarrow \mathbb{R}$ be the Gaussian radial basis function: $$\phi(x) = \exp(-|x|^2)$$ Let $$f_i(x) = \phi{\left(\frac{x - \mu_i}{\sigma_i}\right)}$$ I'm computing the ...
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16 views

Gaussian integral and polar change of variable

I would like to compute the Gaussian integral $\int_{-\infty}^{+\infty}e^{-x^2/2} dx$ using a polar substitution. I know the usual proof using Fubini's theorem and the polar substitution of variables....
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1answer
25 views

integral of 3d gaussian with hollow integral space

I am trying compute the triple integral of a 3D Gaussian within a sphere hollow space. My questions are at the end. You can think the problem in this manner. There is a very large ball whose center ...
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21 views

Two Point Gauss-Laguerre Integration

I have the following question and in the notes we are taught about general gaussian integration and gauss-legendre, but only briefly about gauss-laguerre so I am a bit stuck. The question is as ...
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1answer
35 views

Infinite sum of gaussian exponential

Does anybody know a closed expression for: $$\sum_{n=-\infty}^{+\infty} e^{-(a+bn)^2}$$
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1answer
122 views

Integral over the hypersphere

Assume I have a diagonal matrix $L$ of size $n$. I want to compute the following integral: $$I_n(L) \equiv \int_{(\mathbb{S}^{n-1})^2} \mathrm{d}\sigma(x) \mathrm{d}\sigma(x') \exp[n x^\top L \, x']$$...
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1answer
36 views

Gaussian multi-variate integral

I would like to compute the following integral $$ I_n = \frac{1}{\sqrt{det(2\pi A)}} \int_{\mathbb{R}^n} ||x||^2_2 \exp\left(-\frac{1}{2} x^TAx\right) \mathrm{d} x $$ where $A$ is symmetric and ...
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1answer
29 views

Integrating a 2D Gaussian over a linear strip

How do I show that $$\int_{-\infty}^{\infty} \frac{e^{-\frac{x^2}{2 \sigma ^2}} \left(\text{erf}\left(\frac{2 d-\sqrt{2} x}{2 \sigma }\right)+\text{erf}\left(\frac{2 d+\sqrt{2} x}{2 \sigma }\...
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237 views

Gaussian matrix integration

Consider a random hermitian matrix $B$ of size $N\times N$ with Gaussian probability measure given by $$ dx(B) = e^{-\frac{N}{2}Tr(B^2)}\prod_{i=1}^N dB_{ii} \prod_{i<j} d\Re(dB_{ij})d\Im(dB_{ij}) ...
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1answer
25 views

Expected Value of Gaussian RV conditioned on the summation of two other joint Gaussian RVs

I am trying to solve the following question but I am totally lost on how to approach it. I know the classical division of joint pdf to marginal pdf is indeed the correct answer but I believe there has ...
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1answer
30 views

Approximating the first moment of h(x) where $x$~Lognomal($\mu, \sigma$)

What is the best way to approximate $E(h(X))$, where $X$ ~ Lognomal($\mu, \sigma$). So far, I can think of Monte Carlo Methods and Gaussian Hermite quadrature as below: \begin{align} E(h(X)) &= ...
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Gaussian RV issue

I got a problem in exam which is related to Gaussian RV and is as follows: RV 'W' is a gaussian random variable with $N_w(0,1)$ and Y is a random variable defined as: $$Y = W for |w|<=1 $$ $$Y ...
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1answer
169 views

How to evaluate the integral $\int\mathbf{g}^T\mathbf{v} \exp{(-\frac{1}{2}\mathbf{v}^T \mathbf{A}\mathbf{v})}\,d\mathbf{v}$

I have the following multidimensional Gaussian integral: $$\int\textbf{g}^T\textbf{v} \exp{\left(-\frac{1}{2}\textbf{v}^T \textbf{A}\textbf{v}\right)}\,d\textbf{v},$$ where $\textbf{A}$ is a real ...
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How to find a proper bivariate Gaussian distribution that integrals 95% over a convex polygon

I have a known 2-D convex polygon, and my goal is to find a bivariate Gaussian distribution, that 1) samples from the Gaussian falls into the polygon 95% of the times, and 2) should be reasonably ...
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1answer
28 views

Evaluation of a certain Gaussian integral

I wish to prove that for any $a\in\mathbb R$ $$ \lim_{A\to\infty}\int_{-A}^A \frac{e^{-x^2/2-(a+x)^2/2}(a+2x)}{e^{-x^2/2}+e^{-(a+x)^2/2}}dx = 0. $$ I have verified the above equality for multiple $a\...
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2answers
66 views

Proof inequality $\frac{\sqrt{\pi}}{2}\sqrt{1-e^{-a^2}} < \int_0^a e^{-x^2}dx < \frac{\sqrt{\pi}}{2}\sqrt{1-e^{-2a^2}}$

I'm asked to prove the inequality $$\frac{\sqrt{\pi}}{2}\sqrt{1-e^{-a^2}} < \int_0^a e^{-x^2}dx < \frac{\sqrt{\pi}}{2}\sqrt{1-e^{-2a^2}}$$ After playing around for a while I was able to find (...
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Laplace Method - Estimation of an integral

I am just working on a paper by Shinzo Watanabe on "Asymptotic Evaluations of Wiener Functional expactations": $μ(dx)$ is the $d$-dimensional Gaussian distributions So that is the interesting ...
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41 views

finding area under the curve of a value

I am not a mathematician and I would love if you could explain some things to me, please. I have a data, a list of some values. Minimal value is -3.04, max value is 2.75. I plotted the data and I ...
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1answer
78 views

Derivative of Integral of Gaussian

I'm looking at the Wikipedia page for the derivation of the PDF of the chi-squared distribution, but I'm not sure how they got from the end of the first line to the second line. How do they evaluate ...
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1answer
85 views

Convexity of expectation

Given some function $f$, how could I find out what properties of $f$ make $$F(q) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} f(\sqrt{q} x) e^{-x^2/2} dx$$ either convex or concave? I have some ...
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48 views

Variance of delta function and Gaussian function and their co-variance

For x and y to be statistically independent, I have a probability density function of delta function given as: f(x)= 1/2𝛿(x+1) + 1/2𝛿(x-1), where x is a random variable. From the function, I can ...
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0answers
23 views

'Jacobian' of QR decomposition of a rectangular matrix

I want to calculate the volume of real Stiefel manifold $V_{k}(\mathbb{R}^N)$ . $$ V_{k} (\mathbb{R}^N) = \{ H \in M(N, k, \mathbb{R})| H^{T}H = I_{k} \} $$ ((^T) denotes transposed matrix. $M(N, k, \...
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1answer
38 views

normal distribution pdf sum greater than 1

Not sure where to ask this question, but seems like it is a mathematics confusion. So I am trying produce 1 dimension pdf of normal distribution in c++. The equation I used: $$f(x) = \frac{1}{\sqrt{...
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1answer
33 views

how to solve Gaussian integrals with three easy cases

I am using the the book called street mathematics to learn more about dimensional analysis. I am trying to understand a problem in the book. The question is to use dimensional analysis to find the ...
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1answer
57 views

How to switch to polar coordinates with Gaussian Integral?

I'm trying to do the Gamma function of 3/2, so $$\int_0^{\infty } e^{-x} \sqrt{x} \, dx$$ So far I have this u substitution $$u=\sqrt{x}$$ $$du=\frac{dx}{2 \sqrt{x}}$$ $$\int_0^{\infty } e^{-u^2} u2u \...
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36 views

Euler Poisson integration problem

I wish to calculate the follwoing integral: $$\frac{\intop_c^\infty x^me^{-\frac{1}{2}(x-a)^2}dx}{\intop_0^\infty x^me^{-\frac{1}{2}x^2}dx}$$ for a given $a,c\in\mathbb{R}$ and a given $m\in\mathbb{Z^+...
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1answer
61 views

Trouble proving Gaussian-like integral

I came across Gaussian integrals, and was trying to prove them myself. I proved the basics, but am stuck on the following $$\int\limits_{-\infty}^{\infty}xe^{-a(x-b)^2}dx = b \sqrt{\frac{\pi}{a}}$$ ...
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0answers
10 views

Gaussian integral over the rindler wedge

I'm trying to evaluate analytically, if it's possible, the Gaussian integral over the Rindler wedge, meaning: $\int^{\infty}_{0}dx_1\int_{-x_1}^{x_1}dx_0e^{-b(x_1+a_1)^2}e^{b(x_0+a_0)^2}$ Where $a_1,...
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1answer
15 views

Fail to get original term regarding Multivariate Gaussian Distribution

I have a project regrading Robotics. I want to implement Extended information filter. But today my query is not about Robotics Topic. It is mathematical based doubts. The key formula in which Extended ...