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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 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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An integral of several Gaussian densities

Let $f(x \mid \mu, \sigma)$ be the PDF of the Normal/Gaussian distribution. Is there a way to compute: $$ \int_{-\infty}^{\infty} \frac{f(x \mid \mu, \sigma_1)}{f(x \mid 0, \sigma_1)+f(x \mid 0, \...
Aleksandar Bojchevski's user avatar
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How can I solve this gaussian integral analytically? [closed]

This problem I am getting for my work. A, B, C are some constants. When the integration range is finite how can I solve this? I am having a problem with this integration $$\int_{\omega_a}^{\omega_b} \...
INDRANIL MAITI's user avatar
3 votes
3 answers
155 views

How to evaluate $\int_0^{\infty } e^{-x^2+2 x} \text{erf}(x+1) \, dx$

I am interested in evaluating: $$\int_0^{\infty } e^{-x^2+2 x} \text{erf}(x+1) \, dx$$ By using change of variable $x\to 1+\frac{y}{\sqrt{2}}$, this is transformed into: $$\int_{-\sqrt{2}}^{\infty } \...
mattTheMathLearner's user avatar
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1 answer
31 views

Calculating some Gaussian ratios

Let $N \geq 1$ be a positive integer, and let $w = (w_1, \dots, w_N)$ denote a positive sequence of real numbers. Let $\{g_n\}_{n \leq N}$ denote a sequence of iid standard Normal random variables. ...
Drew Brady's user avatar
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4 votes
2 answers
95 views

Integral of two independent Gaussian with different mean

I am trying to solve the following integral: $$ \int_{-\infty}^{\infty}\int_{-c\left\vert x\right\vert}^{c\left\vert x\right\vert} \frac{1}{2\pi\sigma^{2}}\, \exp\left(-\frac{\left[y - \theta\,\right]^...
Massimiliano Datres's user avatar
1 vote
0 answers
36 views

Multi-mode Gaussian integral with a symmetric complex matrix in the exponent

I am trying to evaluate a multi-dimensional Gaussian integral of the form, \begin{equation} \int_{\mathbb{R}^{n}} d^{n}x e^{-\frac{1}{2}x^{T} A x}, \end{equation} where the matrix $A$ is, Complex ...
cosmicjoke's user avatar
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71 views

Laplace's Approximation for Integral from 0 to Infinity

Background I am working on a problem where I need to evaluate an integral from 0 to infinity and compare the numerical solution to an analytical approximation using Laplace's method. However, I am ...
Alireza's user avatar
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How to show that the Gaussian sequence satisfies the sifting property of the delta function?

I am trying to show that the Gaussian sequence of functions, defined by $g_a (x) = (\frac{1}{a\sqrt{\pi}}) e^\frac{-(x-x_0)^2}{a^2}$ satisfies the sifting property of the $\delta$ - function, namely ...
Falgun Sukhija's user avatar
1 vote
2 answers
61 views

Indefinite Integral Of A Wave Packet

So, I have the following function (also known as a wave packet in physics): $$5{e}^{-\frac{{x}^{2}}{25}} \cos(3x)$$ Now, I want to find the indefinite integral of this and I started with integration ...
Mitsos YT's user avatar
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Expected value of rectified linear function over white noise

I wondered if anyone could handle a rectified linear function's expectation. Looking online, I found this solution, which I assume is correct for one-dimensional $x$. $$ \begin{aligned} & \int_0^{\...
Nosrat Mohammadi's user avatar
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2 answers
60 views

Explicit formula representing the integral $\int_{-\infty}^\infty x^2\Phi(bx+a)\varphi(x)\mathrm d x$

Let $\varphi$ (resp. $\Phi$) be the pdf (resp. cdf) of the standard normal distribution and let $a$ and $a$ be real numbers with $b > 0$. Question. Does the integral $$ \int_{-\infty}^\infty x^2\...
dohmatob's user avatar
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-1 votes
2 answers
138 views

Integral involving erf

I would greatly appreciate if I could get some help in the following integral: $$ \int_{0}^{\infty}\frac{\operatorname{erf}\left(x\right)}{a^{2} +x^{2}}{\rm d}x $$ A similar post was this one. The ...
Sudhendu Ahir's user avatar
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Extension of Dirichlet Divisors in Gauss Circle Integral Point Problem

Dirichlet Divisors A well-known problem in number theory is to study the sum of divisor functions $d(n)$: $D_2(x)=\sum_{n\leq x} d(n)$, so $D_2(x)$ can be expressed as the number of the integer points ...
Skylar swift's user avatar
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How to evaluate these Gaussian integrals?

I am reading this paper by Gross and Mezard. In it they calculate the number of solutions to the Thouless-Anderson-Palmer mean field equations for a spin glass model. As an intermediate step (...
baderi's user avatar
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2 votes
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Using Jacobi-Anger expansion to approximate a Gaussian integral

I am trying to perform the following integral, \begin{equation} \mathcal{I}=\int_{-\infty}^\infty e^{-x^2-ib y \sinh{\left(\frac{x}{b}\right)}}dx \end{equation} with $b$ and $y$ real and $b> 0$. ...
STU's user avatar
  • 117
7 votes
2 answers
295 views

Integral of the product of a Gaussian and a exponential of a hyperbolic function

In a derivation I am working on, I have encountered an integral of the form \begin{equation} \int_{0}^{\infty}e^{-a x^2-b\ \textrm{cosh}(x)}\ dx \end{equation} with $a$ and $b$ real and positive. I am ...
STU's user avatar
  • 117
2 votes
1 answer
73 views

Integral of a rectified power law with Gaussian noise for non-integer powers

I am interested in the following integral: $$f(x)=\int dt \phi(t) [x+\sigma t]_+^n $$ where $\phi(x)=e^{-x^2/2}/\sqrt{2\pi}$ is a standard Gaussian distribution, $[x]_+=max(x,0)$ is a rectification, $...
Uri Cohen's user avatar
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Multivariate gaussian integral with squared quadratic term outside exponent

I have found a post in 1, but can't solve the integral \begin{align} \int(x^{T}Rx)^{2}\exp\big[-\frac{1}{2}(x^{T}Ax)\big]dx, \end{align} with $x \in \mathbb{R}^{n}$, and PD matrices $A,R \in \mathbb{R}...
user1168149's user avatar
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0 answers
46 views

Contourn integration with gaussian-like argument

I'm currently working on evaluating the following integral: $$ I=\int_{-\infty}^{+\infty}\frac{a-x^2+b}{(a-x^2)^2}e^{-i c x^2}dx, $$ where $a,b,c$ are real and positive constants. I wish to compute ...
Albus Black's user avatar
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3 answers
62 views

Evaluate $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}x\,y\,\exp\left({-\frac{1}{2}\left(x-y\right)^2-\frac{y^2}{2}}\right)\,\mathrm{d}x\mathrm{d}y$

I'm currently practicing some problems in probability theory to give me a good base for stochastic processes. Here, probability density for $Y$ is a Gaussian with zero mean and that for $X$ is a ...
Govind's user avatar
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4 votes
2 answers
99 views

Does $ \int_{-\infty}^\infty \int_0^x e^{y^2/2} \, dy \; e^{-x^2/2} \, dx$ exist?

I am trying to figure out if the following integral exists $$ \int_{-\infty}^\infty \int_0^x e^{y^2/2} \, dy \; e^{-x^2/2} \, dx $$ For higher potentials in the exponent of the exponential, say $x^4$, ...
TrippyMushroom95's user avatar
1 vote
1 answer
35 views

Showing a bivariate Gaussian vector has normal components by computing the marginals

Let $( X_1 , X_2 )$ be a bivariate normal random variable. Assume that the correlation coefficient $|\rho[ X_1 , X_2 ] | \ne 1$. Show that $X_1$ and $X_2$ are Normal random variables by calculating ...
Raheel's user avatar
  • 1,711
1 vote
1 answer
29 views

Integral of a product divded by a sum of Gaussian PDFs

Let $f(x)$ and $g(x)$ be the PDFs of two Gaussian distributions both with zero mean, and variances $\sigma_1$ and $\sigma_2$ respectively. I'm trying to compute $$ \int_{-\infty}^{\infty} \frac{f(x) g(...
Aleksandar Bojchevski's user avatar
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1 answer
36 views

Integrating a product of Gaussians by hand

At the end of section 2 of this paper, the authors mention the integral: $$\frac{\alpha}{Z}\int N(x_i | \mu, \sigma I) N(\mu | 0, \rho I) d \mu$$ (Note: $x_i$ and $\mu$ are $d$-dimensional.) In the ...
jeg's user avatar
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4 votes
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107 views

Mean log determinant of a random Gaussian matrix

I wish to calculate the mean log determinant of a random Gaussian matrix, defined by the following definite integral (for $a, b, \sigma\in\mathbb{R}$): $$ I(a, b, \sigma) = \int \frac{d^{n \times n}X}{...
Uri Cohen's user avatar
  • 395
1 vote
1 answer
63 views

Conditions under which the integral of an exponential function over $\mathbb{R}^k$ is finite

Theorem Let $C$ be a normalization constant and $\lambda_1,\ldots, \lambda_k$ a number sequence. The integral over $\mathbb{R}^k$: $$ \int_{\mathbb{R}^k} C \exp\left\{-\frac{1}{2}\sum_{i=1}^k\...
ytnb's user avatar
  • 600
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0 answers
43 views

Radon Transform of Gaussian function

I am trying to find the radon transform of the gaussian function $$f(x,y) = e^{-(x^2 + y^2)}$$ Now, I am using the formula for radon transform as $$ [\mathcal{R}f]{(\rho, \theta)} = \int_{-\infty}^{\...
Subham's user avatar
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1 answer
78 views

$\int_{\mathbb{R}^2} e^{-(x^2+y^2)}=[\int_{\mathbb{R}} e^{-x^2}]^2,$ provided the first of these integrals exists. Munkres Analysis on Manifolds

I am reading "Analysis on Manifolds" by James R. Munkres. (a) Show that $$\int_{\mathbb{R}^2} e^{-(x^2+y^2)}=[\int_{\mathbb{R}} e^{-x^2}]^2,$$ provided the first of these integrals exists. ...
佐武五郎's user avatar
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1 vote
1 answer
62 views

Is there a simple way to evaluate the integral $\int_{\mathbb{R}^n} \delta(x_1^2+\dots+x_n^2-y)\text{d}x_1\dots\text{d}x_n$?

My approach: In general we have: $$\delta(x-x_0)=\frac{1}{2\pi}\int_{\mathbb{R}} e^{-it(x-x_0)}\text{d}t$$ Then ($y \ge 0$): $$I=\int_{\mathbb{R}^n} \delta(x_1^2+\dots+x_n^2-y)\text{d}x_1\dots\text{d}...
Leonardo's user avatar
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1 vote
1 answer
82 views

Non-zero expectation of $p(z)^kz$ for non-even univariate polynomial $p$ with Gaussian variable $z$.

Suppose that $p:\mathbb{R} \rightarrow \mathbb{R}$ is a polynomial and let $z$ be a standard Gaussian variable $(z \sim \mathcal{N}(0,1))$. I am interested in the behavior of the expectation $\mathbb{...
Yatin Dandi's user avatar
1 vote
1 answer
140 views

Gaussian integral with complex coefficients [closed]

We are all familiar with the standard generalized Gaussian integral $$\int_{\mathbb R} dx \ \exp\left(-ax^2+bx\right)=\sqrt\frac{\pi}{a}\exp\left(\frac{b^2}{4a}\right) $$ where $a,b\in \mathbb R$. I ...
Noobgrammer's user avatar
3 votes
2 answers
190 views

Fourier Transform of Gaussian with imaginary pole

I'm struggling at the moment with the following integral, which is essentially an inverse Fourier transform of a Gaussian with a single pole on the imaginary axis: $$I(Q) = \frac{1}{2}\int_{-\infty}^{\...
hedlejo's user avatar
  • 41
0 votes
0 answers
34 views

$\int x^2\exp(-x^2/2\sigma^2)\rho(x)dx\leq C\sigma^2 \int\exp(-x^2/2\sigma^2)\rho(x)dx$

For a 'reasonable' pdf, $\rho(x)$ ($\mathbb{P}(A)=\int_A\rho(x)dx$), I am trying to prove the above inequality. In general, this inequality isn't true since we may take $\rho$ to have support only ...
APP's user avatar
  • 188
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0 answers
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Propagating gaussian wavepacket

I have a problem with evaluating the following gaussian integral in two different ways \begin{equation} \psi(x,t) = \int_{\infty}^{\infty} e^{-\frac{\alpha^2 \hbar^2}{2} (k-k_0)^2} e^{(ikx - i\frac{\...
EM_1's user avatar
  • 259
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0 answers
69 views

Gaussian integral of squared shifted error function

The integral of a shifted error function with respect to the Gaussian measure admits a nice closed form expression: $$ \int_{-\infty}^{\infty}\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{1}{2\sigma^{2}}(x-...
kurtachovo's user avatar
3 votes
0 answers
75 views

$\left(-\frac{1}{2}\right)!$ from the Hypersphere

I was deriving the formula for volume of a n- dimensional Hypersphere when I came across something interesting. It seems like we can define $(0.5)!$ without resorting to the Gamma function. Define: $$...
Tanmay Gupta's user avatar
1 vote
1 answer
121 views

What does $\int_{B_r} h$ mean? Problem 3-41 in "Calculus on Manifolds" by Michael Spivak. $\int_{-\infty}^{\infty}e^{-x^2}dx=\sqrt{\pi}$.

I am reading "Calculus on Manifolds" by Michael Spivak. I solved (a) and (b). Then, I proved $\int_C h=\int_{r_1}^{r_2} \int_{\theta_1}^{\theta_2} rg(r,\theta)d\theta dr$ holds. My question ...
佐武五郎's user avatar
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3 votes
3 answers
119 views

Integrating a product of Gaussian kernels

If we consider the Gaussian kernel $k(x, y) = \exp(-\frac{\Vert x - y \Vert^2}{2\sigma^2})$, then is it true that... $$ \int k(x_i, x)k(x_j, x)dx = k(x_i, x_j) $$ It seems I have seen this before, but ...
Joff's user avatar
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1 vote
0 answers
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Given a multilinear polynomial $g$, and an odd function $f$ show that $\int f(x_i) \exp(-\| x \|^2) g(x-y) d x =0, \forall y,i$ where $g(x)=c$)

Suppose that the following convolution equation holds: for all $y\in \mathbb{R}^n$ and every $i =1 ..n$ $$\int f(x_i) \exp \left(- \frac{\| x \|^2}{2} \right) g(x-y) d x =0$$ where $f(x)$ is an ...
Boby's user avatar
  • 6,015
7 votes
3 answers
238 views

Log-convexity of a function defined by an integral (Normal Mills ratio)

Let's define $f(x)$, for all $x>0$ by : $$f(x)=e^{x^2/2}\int_x^{+\infty}e^{-t^2/2}dt$$ I would like to prove that $f$ is log-convex, which is equivalent to the following condition : $$\forall x>...
Adren's user avatar
  • 7,754
0 votes
0 answers
71 views

Seeking Probability Function Invariant under Normal Gaussian Convolution

I'm currently working on a problem where I need to find a probability function, $P(x)$, that remains unchanged after a normal Gaussian convolution. Specifically, the function should satisfy the ...
Peyman's user avatar
  • 770
1 vote
2 answers
86 views

Computing Gaussian-Weighted Integrals for Small Lengthscales

I am attempting to compute the following integral: $$ I(l) = \int_{[-1,1]^2}dxdy \sqrt{1-x^2} \sqrt{1-y^2}e^{-\frac{(x-y)^2}{l^2}} $$ I am using Mathematica for symbolic calculations and translating ...
user65854's user avatar
3 votes
0 answers
57 views

Generalization of Laplace's integral

A generalization of the Gaussian integral $$ \int_0^{\infty} e^{-\pi at^2} \, dt = \frac{1}{2\sqrt{a}}, \hspace{0.5cm} a>0 $$ is Laplace's integral: $$ \int_0^{\infty} e^{-\pi at^2-\pi b/t^2} \, dt ...
Dave's user avatar
  • 1,763
1 vote
2 answers
216 views

What is the integral $\int_{-\infty}^{+\infty} x^{-2} e^{-x^2}\,dx$?

In my research, I reached to the $$ \int_{-\infty}^{+\infty}\!\! \frac{1}{\sqrt{2 \pi}x^2}e^{-x^2/2}\, dx. $$ I have tried polar change of systems, integral by parts and change of variables but I did ...
Alireza's user avatar
  • 309
3 votes
2 answers
59 views

A partition function problem under conserved total momentum, involving Gaussian integration under a delta constraint

Recently I learned about the following expression showing the kinetic part of the partition function in an N-atom ideal gas under conserved total momentum: \begin{aligned} Q_{\mathrm{Kin}}^{\mathrm{CM}...
Izzy Tse's user avatar
4 votes
1 answer
258 views

Integral over a small set is small and dimension independent

Let $\mu_n$ be the standard normal distribution over $\mathbb{R}^n$. Is the following true? For all $\epsilon > 0$ and $d \in \mathbb{N}$, there exists $\eta > 0$ such that for all $n \geq 1$ ...
Mathews Boban's user avatar
2 votes
0 answers
39 views

Is the function $\int_0^{1} e^{-t H}e^{-2x^2}dt$ in $L^2$ with respect to the gaussian measure?

Consider the function $g(x)=\int_0^{1} e^{-t H}e^{-2x^2}dt$, where $H$ is the differential operator $x\partial_x-\partial_x^2$. Is $g(x)$ in the $L^2$ space with respect to the gaussian measure; i.e. ...
matilda's user avatar
  • 169
1 vote
0 answers
87 views

a special type of Gaussian Integral (reciperocal Quadratic functions with Gaussian Integral)

I have tried to calculate the following integral in terms of $\sigma^2$ : $I=\int_{x=-\infty}^{+\infty}\frac{1}{\sqrt{1+x^2}}e^{-x^2/{2\sigma^2}} dx$ However by using conventional methods such as ...
Alireza's user avatar
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0 votes
0 answers
40 views

If $f \neq 0$, do we have $\int_{ x_i \geq 0} f(x) e^{-\sum_{i=1}^n c_i x_i^2} d^nx \neq 0$ for some $c \in (0,\infty)^n$?

Let $f : \mathbb{R}^n \to \mathbb{R}^m$ be a bounded measurable mapping. Next, define \begin{equation} \mathbb{R}^n_+ := \{ x \in \mathbb{R}^n \mid x_i \geq 0 \text{ for all } i=1,\cdots,n\} \end{...
Keith's user avatar
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0 votes
0 answers
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Extending Laplace transform to multivariate cases?

Let $f : \mathbb{R}^n \to \mathbb{R}^m$ be a measurable $L^\infty$ function. Assume further that $f$ is not zero a.e. and $f(-x)=f(x)$ for a.e. $x \in \mathbb{R}^n$. Then, I wonder if we can conclude ...
Keith's user avatar
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