Linked Questions
12 questions linked to/from Multivariate gaussian integral over positive reals
12
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2
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Gaussian integrals over a half-space
Edit: I shall try to reformulate my question in order to make it -hopefully- more clear.
Let $X$ be a random variable that follows the $n$-dimensional Gaussian distribution. The probability density ...
- 3,086
5
votes
1
answer
1k
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Probability that multi-dimensional random variable is positive?
If we have some multi-dimensional normal probability distribution, $X \sim \mathcal{N}(0,\Sigma)$, with zero mean and a known covariance matrix then what is the probability that every component of $X$...
- 147
11
votes
2
answers
751
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Entropy of fair but correlated coin flips
Consider the joint distribution, $p(\xi_1,...\xi_N)$, with components defined as $\xi_i=\mathrm{sign}(x_i)$, with $(x_1,...,x_N)\sim\mathcal{N}(0,\Sigma)$ with
$
\Sigma_{ij}=\delta_{ij}+(1-\delta_{ij})...
- 185
4
votes
1
answer
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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 ...
- 11.7k
6
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1
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797
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An integral involving error functions and a Gaussian
Let $d\ge 1$ be an integer and let $\vec{A}:=\left\{ A_i \right\}_{i=1}^d$ be real numbers. We consider a following integral:
\begin{equation}
{\mathfrak I}^{(d)}(\vec{A}):=\int\limits_0^\infty e^{-u^...
- 11.7k
10
votes
1
answer
506
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Normal orthant probability in six variables
Let $\mathbf{X} \sim N(\mathbf{0}, \mathbf{\Sigma})$ be a $6$-dimensional Gaussian vector with covariance matrix of the form
$$\mathbf{\Sigma} = \begin{pmatrix} 1 & c \\ c & 1 \end{pmatrix} \...
- 10k
1
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1
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An integral involving a Gaussian, error functions and the Owen's T function.
This question is closely related to An integral involving a Gaussian and an Owen's T function. and An integral involving error functions and a Gaussian .
Let $\nu_1 \ge 1$ and $\nu_2 \ge 1$ be ...
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0
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0
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309
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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_{...
- 638
3
votes
1
answer
147
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Solid Angles Beyond Dimension Three
There is a theorem by Ribando (Measuring Solid Angles Beyond Dimension Three, Discrete Comput Geom 36:479–487 (2006)) https://link.springer.com/content/pdf/10.1007/s00454-006-1253-4.pdf?pdf=button:
...
- 91
1
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0
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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 ...
- 73
0
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0
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Expectation of sign of multivariate gaussian random variables
I wanted find the following expression in closed-form
$\mathbb{E} [sgn(X_1) sgn(X_2) sgn(X_3) sgn(X_4)]$
where each $X_i$ is a gaussian random variable correlated with others. In this regard, we have ...
- 91
1
vote
0
answers
43
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Fourth order statistc of sign function of random vector [duplicate]
we have this preposition as follows
$\textbf{Proposition}$: $\textit{let} \ \boldsymbol{\zeta} \sim
\mathcal{N}(\mathbf{0},{\gamma}\mathbf{I}_N). For \ \mathbf{a}_1 , \mathbf{a}_2 \in \ \mathbb{R}^{N \...
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