# Questions tagged [gaussian]

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### Are the ordinates of the non-trivial zeros of $\zeta(s)$ uniformly distributed around the mid points of Gram point intervals they can be mapped to?

Let $\rho_n$ be the $n$-th non-trivial zero of $\zeta(s)$ and $z_n = \Im(\rho_n)$ with $z_n > 0$ and $z_{n+1} \ge z_n$. A well known method to establish that all $\rho$s reside on the critical line ...
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### Which orthant probabilities are the largest? (For a multivariate normal distribution)

I have a $k$-dimensional multivariate normal distribution $X∼N(0,\Sigma)$ with covariance matrix $\Sigma$. $\Sigma$ has two distinct eigenvalues, say $\lambda_1 > \lambda_2$, with orthogonal ...
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### Hubbard-Stratonovich Transformation

Hello, The Hubbard-Stratonovich transformation $\exp(x^2) = \frac{1}{\sqrt{4 \pi}} \int_{-\infty}^{+\infty} du \exp(-\frac{u^2}{4} - xu)$ allows one to wirte the exponential of a the square of a ...
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### Approximation of integral of gaussian function over a parallelepiped

Remark: I posted this question in math stackexchange here and computer science stackexchange https://cs.stackexchange.com/ few weeks ago but obtain no answer. Given a multi-dimensional gaussian ...
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### Gaussian integrals and Showing $\int f({\vec {x}})e^{\left(-{\frac {1}{2}}\sum \limits _{i,j=1}^{n}A_{ij}x_{i}x_{j}\right)}d^{n}x=e^{D}f|_{x=0}$

This is related to my other question on tackling a gaussian integral for $f(w,u)=\frac{1}{w-u}$. Q1 Suggestions on evaluating gaussian integrals with "nice" functions (not necessarily polynomials) ...
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### when is the average of a function with Gaussian inputs bounded away from zero

Define a function $\phi(x):\mathbb{R}\rightarrow\mathbb{R}$. Consider the expected value function defined as follows \begin{align*} \mu(\beta)=E[g\phi (\beta g)]\quad with \quad g\sim\mathcal{N}(0,1)\...
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### Calculate sample mean confidence interval of noisy logistical distribution

I have $n$ samples which follow a logistic distribution with unknown $u$ and $s$; it is affected by a Gaussian noise with 0 mean. I would like to estimate its average $u$ with a confidence interval (...
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### What is the distribution of determinant of multi multiplication of some Gaussian matrices?

I have a square matric $H = (ABC)(ABC)^H$ where $A$ and $C$ are complex Gaussian matrices with some correlation matrices and $B$ is a diagonal matrix with entries $e^{j \theta}$ on the diagonal such ...
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### Law of OU process with time-dependent dynamics

Fix a non-negative integer $k$ and let $M^1:\mathbb{R}^n\rightarrow \mathbb{R}^n$ and $M^2,\Sigma:\mathbb{R}^n \rightarrow \mathbb{R}^{n\times n}$ be $k$-times continuously differentiable functions, ...
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### Gaussian order statistics

Setup. Let $\alpha\in(0,1)$ fixed; and $\tau\in[0,1]$ (think of it very close to one). Suppose $X_1,\dots,X_n$ are i.i.d. standard normal. Let $Y_1,\dots,Y_n$ be another sequence of standard normals ...
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The $n$-th Gauss-Laguerre quadrature scheme aims to approximate integral of exponentially decreassing function over $[0;+ \infty[$ by a finite sum, according to: $\displaystyle { \int _0 ^{+ \infty} ... 0answers 30 views ### How to characterize the variance of a linear Gaussian system with switching? Consider a random process described by the following linear dynamics: $$x_{k+1} = a x_k + n_k,$$ where$|a|<1$and$n_k$s are i.i.d. standard normal distributed. It is quite easy to prove that$...
I want to prove the following relation for $X\sim \mathcal{N}(0,1)$, $x\in \mathbb{R}$ and $f(x)=\mathbb{E}[\max(X,x)]$: $$f(\frac{f(x+1)+f(x-1)}{2})\leq \frac{f(f(x)-1)+f(f(x)+1)}{2}$$ It seems that ...