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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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Gaussian measure of intersection of halfspaces

Let $a_{1}, a_{2}\in\mathbb{R}^{d}$, $X\sim\mathcal{N}(0,\Sigma)$ be a Gaussian random vector in $\mathbb{R}^{d}$ and let $p$ denote its density. Consider the following integral \begin{equation} \int_{...
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Kalman filter: the bayesian approach derivation some clarifications

I'm reading the book Methods and algorithms for signal processing from Moon Stirling at page 592 there is a derivation of Kalman filter using the Bayesian approach. I have some issues in understanding ...
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Can we calculate gaussian integrals using non-typical norms on R^2

We might know the trick to calculate a gaussian integral $G_2 = \int_{-\infty}^{\infty} e^{-x^2}\,\mathrm{d}x = \sqrt{\pi}$ by instead computing $G^2_2$ and taking advantage of the 2D-plane: $G_2^2 ...
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Limits and Jacobian for Gaussian Integrals

Ahoy everyone! I am new to Gaussian Integrals and my teachers cannot help me out (because they don't get it). So I turn to the Internet for answers. I have very basic doubts and would really ...
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Help with dealing with “basic” linear algebra in a Gaussian path integral

This is my first post in math.stackexchange. This is my Problem: I'm trying to solve the following Gaussian path integral: \begin{equation} I=\lim\limits_{N \to \infty}\int \left(\prod_{n=1}^{N-1} \...
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70 views

Evaluation of the integral using the Laplace approximation?

How can I evaluate the following integral using the Laplace approximation at any given point $x$? \begin{equation} x \mapsto \int \sigma(w^T x) \mathcal{N}(w; 0, \Sigma) dw \,, \end{equation} ...
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1answer
21 views

For a Fixed Variance, Gaussian Distribution Maximizes Entropy?

I was reading this paper. In page 5, second column, they mention that, $$h(Q) + h(P) \ge log(e \pi) => \sigma(Q) * \sigma(P) >= \frac{1}{2}$$ Where entropy $h$ is defined in the following way:...
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Am I using the Simpsons rule and Gauss-Legendre method correctly? [closed]

I have the integral here: Simpsons rule: Answer 414.11411 Gauss-Legendre method Here the limits are x+y I found the answer to be around 0.923 Just wanted to make sure these values are correct.
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Can every Gaussian integral be reduced to elementary functions and poly-logarithms only?

Let us define a following function: \begin{eqnarray} {\mathcal J}^{(d)}(\vec{A}) := \int\limits_0^\infty e^{-u^2} \prod\limits_{\xi=1}^d erf(A_\xi u) du \end{eqnarray} for $\vec{A}:=\left(A_\xi\right)...
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Variance of truncated 2d Gaussian

To find the expectation of the Truncated Gaussian E($z_1^2$| $z_1^2 \leq \tau , z_2^2 \geq \tau$). Where $\boldsymbol{z} = [z_1,z_2]^T$ and $\boldsymbol{z} \sim \mathcal{N}(\boldsymbol{0},C)$, where $...
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Two independent normal random variable probability question

Two independent random standard normal variable X and Y, find the probability that: $P(X>2Y)$ and $P(X>2|Y|)$ I'm pretty sure the first one is $0.5$, because $X-2Y$ should also be a normal ...
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Calculate a 2-dimensional Gauss-Hermite quadrature approximation

I need to calculate an integral in which the second term is a bivariate normal density. I thought about using a 2-dimensional Gauss-Hermite quadrature. But not being familiar with the subject, I do ...
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Approximating an unknown probability with a Gaussian and then combining these probabilites?

I have a candidate C with unknown features $F = [F_1, F_2, … , F_n]$ where $F_i$ is the probability that the candidate has the feature , unfortunately this is poorly know and must be approximated so ...
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Why are isolated zeros allowed in the weight function?

In an exercise I have determined the Gaussian Quadrature formula and I have applied that also for a specific function. Then there is the following question: Explain why isolated zeros are allowed ...
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Any good approximation for $\Phi^{-1}(1-\Phi(a))$?

Let $\Phi$ be the standard Gaussian CDF and $a > 0$. Question Is there any good approximation for $\Phi^{-1}(1-\Phi(a))$ ?
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Distribution of Gaussian Random variable. Concentrated measure.

Let $\mu$ be the standard Gaussian distribution on $\mathbb{R}$. Show that if $B$ is a Borel set (w.r.t the Euclidean metric) and $\mu(B)\geq 1/2$ then $$\mu(B_{r})\geq 1-\frac{1}{2}e^{-\frac{t^2}{2}...
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72 views

Generalized Owen's T function

As Wikipedia teaches us https://en.wikipedia.org/wiki/Owen%27s_T_function the Owen's T function $T(h,a)$ defines a probability of a bivariate event $X>h$ and $0<Y<a X$ where $X,Y$ are ...
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Mean estimator of a Gaussian variable with positive mean for quadratic loss

Suppose $\phi, \Phi$ are PDF and CDF for a $1$-dimensional normal Gaussian, and $X\sim\mathcal{N}(\theta,1)$, in which $\theta>0$ is positive but othrewise unknown. We want to estimate $\theta$ ...
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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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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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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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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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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
29 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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Asking for a function name [closed]

Is there any name for the function $z=\exp(-(x^2+y^2))$?
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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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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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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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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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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
56 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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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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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
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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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27 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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38 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
160 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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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
44 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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245 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
30 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
31 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)) &= ...