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.
888
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Evaluation of Gaussian integral $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx$
How to prove
$$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}$$
- 8,008
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votes
3
answers
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reference for multidimensional gaussian integral
I was reading on Wikipedia in this article about the n-dimensional and functional generalization of the Gaussian integral. In particular, I would like to understand how the following equations are ...
- 8,750
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votes
1
answer
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Why do zeta regularization and path integrals agree on functional determinants?
When looking up the functional determinant on Wikipedia, a reader is treated to two possible definitions of the functional determinant, and their agreement is trivial in finite dimensions.
The first ...
- 152k
20
votes
2
answers
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Which functions are tempered distributions?
Today's problem originates in this conversation with Willie Wong about the Fourier transform of a Gaussian function
$$g_{\sigma}(x)=e^{-\sigma \lvert x \rvert^2},\quad x \in \mathbb{R}^n;$$
where $\...
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votes
1
answer
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Entropy of the multivariate Gaussian
Show that the entropy of the multivariate Gaussian $N(x|\mu,\Sigma)$ is given by
\begin{align}
H[x] = \frac12\ln|\Sigma| + \frac{D}{2}(1 + \ln(2\pi))
\end{align}
where $D$ is the dimensionality of ...
- 1,133
13
votes
5
answers
384
views
How to show $ \int_{-\infty}^{\infty} \frac{e^{-(x+1)^2}}{1+e^{-x}}\mathrm{d}x = \frac{\left(2\sqrt[4]{e} -1 \right)\sqrt{\pi}}{2e}$?
I was recently looking at this post where the following formula is shown:
$$
\int_{-\infty}^{\infty} \frac{E(x)}{1+\mathcal{E}(x)^{O(x)}}\mathrm{d}x= \int_0^{\infty} E(x) \mathrm{d}x
$$
where $E(x), \...
- 7,233
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votes
3
answers
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Arnold Trivium Problem 39
We find in Arnold's Trivium the following problem, numbered 39. (The double integral should have a circle through it, but the command /oiint does not work here.)
Calculate the Gauss integral
$$...
- 40.2k
12
votes
4
answers
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Why can $\int_{x=0}^{\infty} x\,\mathrm{e}^{-\alpha x^2}\mathrm {d}x$ not be evaluated by parts to obtain $\frac{1}{2\alpha}$?
Can $\int_{x=0}^{\infty} x\mathrm{e}^{-\alpha x^2}\,\mathrm {d}x$ be evaluated by parts to show that $\int_{x=0}^{\infty} x\mathrm{e}^{-\alpha x^2}\,\mathrm {d}x= \frac{1}{2\alpha}$
I know that this ...
- 8,468
12
votes
1
answer
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Show that integral of Gaussian distribution is 1
Under a normal distribution, μ = 0 and σ = 1, but when then integrating this equation, I get an error function.
Without using Riemann sums, how can I prove that this equation = 1?
I have only had a ...
- 123
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votes
2
answers
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Is $g(x)= \frac{x_1^2}{r_1^2}+\frac{x_2^2}{r_2^2}- c$ a unique solution of $E[g(Z)\mid M=\mu]=0, \forall \mu \in \text{ellipse } C$ for $Z$ Gaussian
Let $Z \in \mathbb{R}^2$ be an i.i.d. Gaussian vector with mean $M$ where $P_{Z\mid M}$ is its distribution.
Let $g: \mathbb{R}^2 \to \mathbb{R}$ and consider the following equation:
$$
E[g(Z)\mid M=\...
- 5,985
11
votes
3
answers
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Compute multivariate complex Gaussian integral
I don't know how to work out the homework of Leib&Loss P121, Ex4(b), in which we need to compute the following
$$
\int_{\mathbb{R}^n}\exp(-x^tAx)dx=\pi^{n/2}/\sqrt{\det A}
$$
where $A=A^t$ is a ...
- 1,461
10
votes
1
answer
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Integrating:$\int\limits_0 ^ {\infty}e^{-x^2}\ln(x)dx $
$$ \displaystyle \int_0 ^ {\infty}e^{-x^2}\ln(x)dx = -\frac{1}{4}(\gamma +2\ln(2))\sqrt{\pi} $$
This is a well known integral.
But I want to know how to solve it??
Also, please refrain using contour ...
- 447
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3
answers
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gaussian integral of power of cdf : $\int_{-\infty}^{+\infty} \Phi(x)^n \cdot \phi(a+bx) \cdot dx$
Is there an analytic solution for the following Gaussian integral?
$$\int_{-\infty}^{+\infty} \Phi(x)^n \cdot \phi(a+bx) \cdot dx$$
with
$n$, a positive integer (typically under 10)
$a,b$, real ...
- 101
10
votes
1
answer
370
views
Intuition for $N(\mu, \sigma^2)$ in terms of its infinite expansion
To gain deeper insight to the Poisson and exponential random variables, I found that I could derive the random variables as follows:
I consider an experiment which consists of a continuum of trials ...
- 704
10
votes
2
answers
951
views
Tighter tail bounds for subgaussian random variables
Let $X$ be a random variable on $\mathbb{R}$ satisfying $\mathbb{E}\left[e^{tX}\right] \leq e^{t^2/2}$ for all $t \in \mathbb{R}$. What is the best explicit upper bound we can give on $\mathbb{P}[X \...
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votes
1
answer
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Evaluating the integral $ \int_0^1 \frac{e^{-y^2(1+v^2)}}{(1+v^2)^n}dv$
I am trying to evaluate the integral
$$
\int_0^1 \frac{e^{-y^2(1+v^2)}}{(1+v^2)^n}dv = e^{-y^2}\int_0^1 \frac{e^{-y^2v^2}}{(1+v^2)^n}dv
$$
for $n\in \mathbb{N}$.For n=1 one finds Owen's T function, i....
- 472
10
votes
0
answers
370
views
Multivariable Integral, How to compute it?
Q
How to evaluate a multivariate integral with a Gaussian weight function?
$$
\mathcal{Z_{n}}
\equiv\int_{-\infty}^{\infty}
\exp\left(-a\sum_{j = 1}^{n}x_{j}^2\right)\,
{\rm f}\left(x_{1},x_{2},\...
- 479
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votes
1
answer
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Fourier transform of Hermite polynomial times a Gaussian
What is the Fourier transform of an (n-th order Hermite polynomial multiplied by a Gaussian)?
- 309
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votes
1
answer
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What is the expected value of cosine of a multivariate Gaussian?
Suppose $X \sim \mathcal{N}\left(\mu, \Sigma\right)$. How do I evaluate $\operatorname{E}\left[\cos \left(t^{T}X \right) \right] $ and $\operatorname{E}\left[\sin \left(t^{T}X\right) \right] $? Does ...
- 993
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votes
1
answer
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views
Is $\int_{M_{n}(\mathbb{R})} e^{-A^{2}}d\mu$ a convergent integral?
Is the following integral a convergent integral? Can we compute it, precisely?
$$\int_{M_{n}(\mathbb{R})} e^{-A^{2}}d\mu $$
Here $\mu$ is the usual measure of $M_{n}(\mathbb{R})\simeq \mathbb{R}^{n^...
- 848
9
votes
3
answers
655
views
Does $\operatorname{div}\left(\nabla G +xG\right)=0\Longleftrightarrow \nabla G +xG=0$?
Let $G$ be a smooth function defined on $\textbf{R}^d$. What are the assumptions I should use to assume that
$$\operatorname{div}\left(\nabla G(x) +xG(x)\right)=0 \quad (\forall x\in \textbf{R}^d)$$
...
- 1,528
9
votes
2
answers
730
views
Gaussian matrix integration
Consider a random hermitian matrix $B$ of size $N\times N$ with Gaussian probability measure given by
$$
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})
...
- 9,878
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votes
7
answers
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How to integrate $\int_{-\infty}^\infty e^{- \frac{1}{2} ax^2 } x^{2n}dx$
How can I approach this integral? ($0<a \in \mathbb{R}$ and $n \in \mathbb{N}$)
$$\large\int_{-\infty}^\infty e^{- \frac{1}{2} ax^2 } x^{2n}\, dx$$
Integration by parts doesn't seem to make it ...
- 4,542
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votes
4
answers
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What is the purpose of $\frac{1}{\sigma \sqrt{2 \pi}}$ in $\frac{1}{\sigma \sqrt{2 \pi}}e^{\frac{(-(x - \mu ))^2}{2\sigma ^2}}$?
I have been studying the probability density function...
$$\frac{1}{\sigma \sqrt{2 \pi}}e^{\frac{(-(x - \mu ))^2}{2\sigma ^2}}$$
For now I remove the constant, and using the following proof, I prove ...
- 819
8
votes
4
answers
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Gaussian type integral $\int_{-\infty}^{\infty} \frac{\mathrm{e}^{-a^2 x^2}}{1 + x^2} \mathrm{d}x$
When working a proof, I reached an expression similar to this:
$$\int_{-\infty}^{\infty} \frac{\mathrm{e}^{-a^2 x^2}}{1 + x^2} \mathrm{d}x$$
I've tried the following:
1. I tried squaring and ...
- 1,252
8
votes
3
answers
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views
Evaluate $\int_{0}^{\infty} \mathrm{e}^{-x^2-x^{-2}}\, dx$
I have to find
$$I=\int_{0}^{\infty} \mathrm{e}^{-x^2-x^{-2}}\, dx $$
I think we could use
$$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2} $$ But I don't know how.
Thanks.
8
votes
2
answers
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Higher Order Terms in Stirling's Approximation
Some websites and books give stirling approximation as
$$n! = \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n \left( 1 + O \left(\frac{1}{n} \right)\right)$$
However when I check their derivations most ...
- 309
8
votes
4
answers
360
views
Finding the anti-derivative of $ \frac{e^{-c y^2 }}{y\sqrt{y^2-1}}$
I am trying to evaluate the integral
\begin{align}
\frac{1}{2\sqrt{2}\pi}\int_{0^{-}}^{t} ds \ \frac{e^{-x^2/2S^2(t,s) }}{\Sigma(s) S(t,s)}
\end{align}
where $S(t, s) = 2D(t-s)+\frac{\Sigma(s)}{2}$ ...
- 472
8
votes
1
answer
164
views
Is $\int_{M_{n}(\mathbb{R})} e^{-A^{2}}d\mu$ a convergent integral?(2)
We identify $M_{n}(\mathbb{R})$ with $\mathbb{R}^{n^{2}}$
We put $\int_{M_{n}(\mathbb{R})} e^{-A^{2}}d\mu=\lim_{r\to \infty} \int_{D_{r}} e^{-A^{2}}$ where the later is counted as a Riemann ...
- 848
8
votes
0
answers
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views
Approximation of integral of gaussian function over a parallelepiped
Given a multi-dimensional gaussian function, defined by
$$f(\boldsymbol{x})=\exp\left\{-\frac{1}{2} \boldsymbol{x}^T\boldsymbol{x} \right\}=\exp\left\{-\frac{1}{2} \sum_{i=1}^nx_i^2 \right\}$$
And an ...
- 16k
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votes
1
answer
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Gaussian Integral using contour integration with a parallelogram contour
I'm having trouble figuring out how to use contour integration to compute the Gaussian integral. The contour I'm using is a parallelogram with function,
$f(z) = \Large \frac{ e^{i \pi z^2}}{sin(\pi z)}...
7
votes
3
answers
228
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 :
$$\forall x>...
- 7,535
7
votes
2
answers
986
views
Gaussian Integral using single integration
So the Gaussian integral basically states that:
$$ I = \int_{-\infty}^{\infty} e^{-x^2} \ dx =\sqrt{\pi}$$
So the way to solve this is by converting to polar co-ordinates and doing a double ...
- 71
7
votes
1
answer
763
views
How to evaluate $\displaystyle\int_{-\infty}^{\infty}e^{-\frac{x^{2}}{2}}\ln\left(1+e^{x}\right)\mathrm{d}x\textbf{ efficiently}$?
I am struggling in evaluating the follow integral:
$$\int_{-\infty}^{\infty}e^{-\frac{x^{2}}{2}}\ln\left(1+e^{x}\right)\mathrm{d}x\overset{\text{Wolfram}}{\approx} 2.02049$$
$\color{red}{\text{Despite ...
7
votes
1
answer
597
views
Covariance of a rectified (relu) Gaussian
Given a normal random vector $$X\sim N(\mu,\Sigma)$$ for spd $\Sigma$, I'm interested in the covariance matrix $K=\mathrm{cov}(Y)$ of the variable $$Y = \mathrm{relu}(X)$$ where the relu is performed ...
- 141
7
votes
1
answer
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The integral $\int_{-\infty}^\infty \frac{e^{-x^2}}{(1+e^x)^2} dx$.
Let
$$T(n) = \int_{-\infty}^\infty \frac{e^{-x^2}}{(1+e^{x})^n} dx.$$
We have that
$$ T(0) = \sqrt{\pi} \text{ and } T(1) = \frac{\sqrt{\pi}}{2}$$
and also that
$$ T(3) = \tfrac{3}{2} T(2) - \frac{\...
- 145
7
votes
2
answers
197
views
Evaluating $ \int_{-\infty}^{t} e^{-(\tau+a)^2} \mathrm{erf}(\tau) \mathrm{d}\tau$
I need to evaluate this integral:
$$
I(t,a) = \int_{-\infty}^{t} e^{-(\tau+a)^2} \mathrm{erf}(\tau) \ \mathrm{d}\tau
$$
where the $\mathrm{erf}(\tau)$ is the error function.
I can prove that this ...
- 103
6
votes
3
answers
164
views
How do we evaluate the closed form for $\int_{-\infty}^{+\infty}{(-1)^{n+1}x^{2n}+2n+1\over (1+x^2)^2}\cdot e^{-x^2}\mathrm dx?$
Proposed:
$$\int_{-\infty}^{+\infty}{(-1)^{n+1}x^{2n}+2n+1\over (1+x^2)^2}\cdot e^{-x^2}\mathrm dx={\sqrt{\pi}\over 2^{n-2}}\cdot F(n)\tag1$$
Where is n integer, $n\ge1$
I am struggled to find ...
- 8,090
6
votes
2
answers
1k
views
Fourier transform of squared Gaussian Hermite polynomial
I'm attempting to calculate the first-order perturbation energy shift for the quantum harmonic oscillator with a perturbing potential of $V(x)=A\cos(kx)$. Omitting the relevant physical factors, I've ...
6
votes
3
answers
11k
views
Computing the integral $\int \exp(ix^2) dx$
I'm trying to compute the following integral: $I_1 = \int_{-\infty}^\infty \exp(iu^2) du$.
This is what I did, but it is wrong and I don't know why:
$$I_1^2 = \left (\int_{-\infty}^\infty \exp(iu^...
- 85
6
votes
2
answers
628
views
Solve: $\int_{0}^{\infty}{ \left( \cos \left( ... \right)+\sin \left( ... \right) \right) \frac{e^{-u^2}}{(u^2+ \frac{\alpha}{4t})^2} }~\mathrm{d}u$
I have a really nasty integral to solve as follow: (It is good for challenge lovers!)
$$
I(t)=\int_{0}^{t}{{\bigg(\cos\left({\frac{\gamma}{4(t-r)}}\right)+\sin\left({\frac{\gamma}{4(t-r)}}\right)\...
- 407
6
votes
2
answers
458
views
Gaussian-trigonometric definite integral $\int_0^\infty \frac{e^{-x^2}}{1+a \cos x}dx$
Is it possible to evaluate this integral in closed form?
$$I(a)=\int_0^\infty \frac{e^{-x^2}}{1+a \cos x}dx$$
$$0<a<1$$
I encountered this integral when trying to find a closed form for the ...
- 31.5k
6
votes
1
answer
483
views
Polar Coordinate Transformation - Motivation
I am trying to work out the reason why the integral
$$ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{e^{-(x^2+y^2)}}\,dx\,dy $$
is, in polar coordinates,
$$ \int_{-\infty}^{\infty}{e^{-r^2}} \,r\,...
- 1,059
6
votes
1
answer
170
views
A certain Gaussian integral
Can somebody evaluate the following Gaussian integral?
$$
I(t,\sigma) := \int_{-\infty}^\infty \frac{dx e^{-x^2/(2\sigma^2)}}{\sqrt{2\pi \sigma^2}}
\frac{\sin{\left(2 t\sqrt{1+x^2} \right )}}{\sqrt{1+...
- 2,506
6
votes
2
answers
1k
views
Integral involving erf and exponential
Problem
I would like to compute the integral:
\begin{align}
\int_{0}^{+\infty} \text{erf}(ax+b) \exp(-(cx+d)^2) dx \tag{1}
\end{align}
I have been looking at this popular table of integral of the ...
- 61
6
votes
1
answer
1k
views
Polar coordinates for a gaussian random vector
I've been finding some prolems in solving exercise 3.3.7 of Vershynin's book "High dimensional probability".
Let $X\sim N(0,I_n)$ standard multivariate normal random vector on $\mathbb{R}^n$ ...
- 664
6
votes
1
answer
701
views
Representing the determinant of a Hermitian matrix as an integral
Let $M=\left (\omega\mathbb{I}-A\right )\left(\omega^{*}\mathbb{I}-A^{\dagger}\right)$ be a Hermitian matrix of size $n\times n$ where $A$ is a real non symmetric matrix and $\omega=a+\mathrm{i}b$. $A^...
- 115
6
votes
1
answer
2k
views
Integral over the hypersphere
Assume I have a diagonal matrix $L$ of size $n$. I want to compute the following integral:
$$I_n(L) \equiv \int_{(\mathbb{S}^{n-1})^2} \mathrm{d}\sigma(x) \mathrm{d}\sigma(x') \exp[n x^\top L \, x']$$...
- 402
6
votes
1
answer
2k
views
Integral of Multivariable Gaussian across a Circular Domain.
Basically, trying to compute the probability, a.k.a volume under the bi-variate gaussian distribution: $$ f(x,y) = \frac{1}{2 \pi \sigma^2} \cdot \exp(\frac{-x^2 - y^2}{2 \sigma^2}) $$ over axis-...
- 61
6
votes
0
answers
249
views
Improvement of Jensen inequality for random variables
Jensen inequality implies that for every real random variable $X$ and every integer $n\in \mathbb N$
$$ (\mathbb E[X^2])^n \,\leq\, \mathbb E[X^{2n}]$$
by convexity of the function $x\mapsto x^n$ for $...
- 893