# Questions tagged [gaussian-integral]

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

432 questions
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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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### 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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### 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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### 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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### 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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### 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}) ...
### 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)) &= ...