Questions tagged [gaussian]

For questions about the Gaussian probability distribution, its definition, properties and use.

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134 views

Fourier transform of $e^{-\frac{1}{2}x^2}$ [duplicate]

Fourier transform of $$e^{-\frac{1}{2}x^2}$$ I know to use the Fourier transform direct formula however I keep getting an algebraic mess. so if someone could help me out with detailed steps that ...
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15 views

How to see that the Wick product has $0$ expectation.

In the book "Gaussian Hilbert Spaces" (Svante Janson) the author introduces the Wick product of a finite sequence of $n$ random variables living in a Gaussian Hilbert space $G$ as the orthonormal ...
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1answer
15 views

Simple upper bound for $t(2\Phi(\alpha/t) - 1)$, where $\alpha > 0$ and $t \in (0, 1)$

Let $\alpha > 0$ and $ t \in (0, 1)$. For simplicity, take $\alpha=1$. Let $\Phi$ be the normal cumulative distribution function. Of course, the core of the problem is the term $t\Phi(\alpha/t)$. ...
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8 views

Finding a new lobe in the product of 2 anisotropic Gaussians

Fisher Bingham (Isotropic) case For my current research I am trying to find the parameters of the product of 2 ASG (anisotropic spherical Gaussians). A common spherical Gaussian formula (SG) is: $...
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29 views

Kernel Functions Generating Nonnegative Definite Matrices

Setup I'm studying Gaussian process regression, whose theory comes from stochastic processes and probability, and came across a result I'm not sure how to prove formally. Many people handwave over ...
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5 views

Decomposition of Gaussian spaces with respect to covariance function

Let $K(t,s):T^2 \to \mathbb{R}$ be a kernel symmetric and type positive (for every $n$ $\sum^n_{i,j}u_iu_jK(t_i,t_j) \geq 0$ and $(u_1,\dots,u_n) \in \mathbb{R}^n$) where $T$ is any set. Thus, it is ...
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4 views

Tail Bound for Squared Noncentered Gaussian

I am trying to upper bound the event $$P((x-\lambda)^2 < c)$$ where $x \sim N(\mu, 1)$, $c > 0$, and $\lambda \in [0, 1]$. While I am aware of chi-squared tail bounds for standard squared ...
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10 views

Limit of the hitting time of a Gausian process

Let $(Y_t)$ be a Gaussian process and $\tau_n:=\inf\lbrace t>0:Y_t=n\rbrace$. Does anyone have an idea how I can show that $\lim\limits_{n\to\infty}\tau_n=\infty$ almost surely?
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8 views

Implement an algorithm that samples from the joint posterior (Heteroscedastic Gaussian Processing regression)

I was trying to solve this algorithm but I failed, so any help on this matter is welcome. Consider the following heteroscedastic GP regression, $y_i = f(x_i) + \...
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1answer
15 views

Developing a distribution and multiplying by random envelope

I realize this does not make sense what I'm trying to do below *. So I am rephrasing: I have data that takes on values from [-1,1] heavily centred around zero, say distributed Gaussian about 0. I ...
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13 views

Maxima of isonormal Gaussian process

Let $\mathcal{H}$ denote a real separable Hilbert space and let $W\colon \mathcal{H} \to \mathbf{R}$ denote an isonormal Gaussian process; that is $\mathbf{E}[W(t)W(s)] = \langle t, s \rangle_{\...
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25 views

Anti-concentration for Gaussian

Is there a reasonable anti-concentration bound for Gaussian? Let $X\sim\mathcal N(0, \sigma^2)$, can we get $P(|X|>\epsilon)>1-\delta$? Thanks.
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Convolution of a gaussian noise vector with a sinusoid?

I'm trying to understand what a convolution operation does. Right now, I can see that convolution output is high when the signal contains certain frequencies. For example, $f(x) = sin(wx)$ and if I ...
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14 views

Can someone explain why we have this for a GP regression conditioned on the observations

Consider a Gaussian Process (GP) regression $y_t=f(x_t)+\epsilon_t$ with iid noise $\epsilon_t \sim N(0,\sigma^2)$. Can someone please explain why conditioned on $(y_1, \ldots, y_{t-1})$, $\{x_1, \...
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20 views

Tail bound of normally distributed variable by a exponential

I have $z \sim N(0,n)$ By my script, a random variable $\xi \sim N(0,1)$ satisfies the following tail bound for all $t \ge 0$, $$P(\xi \ge 0) \le e^{-\frac{t^2}{2}}$$ Goal of the derivation is to ...
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1answer
17 views

Estimation of squared normal distribution

I am given a $w \sim N(0,I_n)$ and $w \in \mathbb{R}^n$ and $X \in \mathbb{R}^{n \times d}$ such that $X_1,..., X_d \in \mathbb{R}^n $ of $X$ that satisfy $\|X_i\|^2 = n$ where $n$ is a scalar and ...
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1answer
16 views

Simple function that returns a Gaussian curve?

I need a simple function that it's output is a Gaussian for $\mathbb{R} \to [0,1]$. Any tips? Thanks.
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1answer
25 views

Calculating the mean and standard deviation of a Gaussian mixture model of two curves

An ELO rating is a Gaussian curve with a mean and a standard deviation. Assuming there are two such ratings that belong to the same player (he's using two separate online identities so he has two ...
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6 views

Computation of conditional probability distribution in Gaussian Process Regression

TL;DR: Why is $p(\textbf{y}|\textbf{f}) = \mathcal{N}(\textbf{f}, \sigma^2_{noise}I)$ in p1941 of the paper on Sparse Approximate Gaussian Process Regression here? Question w/ details: Let training ...
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2 views

Gaussian process for surrogate model

I am studying for surrogate modeling approaches that can be used for sensitivity analysis. It seems that Gaussian process is one of the main approaches for building surrogate model. Why Gaussian ...
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7 views

How do we theoretically analyze function spaces over mixed inputs (discrete and continuous input variables)?

I want to analyze "similarity" structures over mixed input spaces (discrete and continuous). My question is whether there exists a natural space over which we can analyze functions (for e.g., ones ...
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6 views

Small gaussian width implies large polar body gaussian measure

This is Ex. 6.14 from http://math.univ-lyon1.fr/~aubrun/ABMB/ABMB.pdf. First some definitions. For a set $L$ let $w(L) = \mathbb{E} \max_{x \in L} \langle x, g \rangle$ where $g \sim N(0, I)$. Also ...
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Isonormal Gaussian process associated with a Hilbert space.

We consider the isonormal Gaussian process $W=\{W(h),h\in H\}$ indexed by a separable Hilbert space $H$, defined on a complete probability space $(\Omega, \mathcal F,P)$ where $\mathcal F:=\sigma(W)$,...
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1answer
39 views

Likelihood Function of Normal Variables

We know that for $i.i.d.$ random variable $Y \sim N(\mu, \sigma^2)$, $Var(Y) = \frac{1}{n}\sum_{i = 1}^n (y_i - \mu)^2$, and the likelihood function for $Y_1, Y_2, ..., Y_n$ is $$l(\mu, \sigma|Y) \...
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21 views

Approximate Beta density function to Gaussian density

Any coefficient, $\beta_0 \in [\beta_L, \beta_U]$ is modeled in the Beta distribution and its pdf is given by $$p(\beta_0) = \frac{\Gamma(\lambda_1+\lambda_2)}{(\beta_U-\beta_L)\Gamma(\lambda_1)\...
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29 views

Sum of random variables $\mathbf{x}\sim \text{Uniform}(\mathbb{S}^{n−1})$ converges to Gaussian?

Let $\mathbf{x}=(x_1,...,x_n)\sim \text{Uniform}(\mathbb{S}^{n-1})$ and $z\sim\mathcal{N}(0,1)$. How to prove for any $A\subseteq \mathbb{R}$ $$\Big|\mathbb{P}\Big\{\sum_i x_i \in A \Big\} - \mathbb{P}...
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8 views

How to get a general Gaussian function according to the constrain conditions

We know the base formula $ y=e^{x^2} $, so how could I get an general form $ y=h_0e^{\frac{(x-b)^2}{2c^2}}+d $ which is settled for conditions below by translating and scaling? $$ y(0)=0 $$ $$ y(L)=-...
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1answer
79 views

Integral of product of 3D Gaussians

I am reading a paper, Keep it SMPL: Automatic Estimation of 3D Human Pose and Shape from a Single Image, CVPR 2016, that models the human parts with capsules for estimating the interpenetration ...
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24 views

Distribution of the entries of a particular matrix product

Let us assume a Complex Gaussian i.i.d. matrix $A$ which can be decomposed using the SVD into $A=UDV^*$, where $U$ and $V$ contain the left and right singular vectors, respectively, and $D$ is a ...
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21 views

About the ERF function (Characteristic Function of the Gaussian Distribution)

Can someone explain to me why that is an $\frac{1}{2}$ multiplying the result of this integral (I think I can understand the rest. I did the calculations, but can't find this 1/2): $$ \int \frac{1}{\...
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16 views

demonstration about a particular triple integral

i have the following integral $L=\int_{-\infty }^{\infty}(\int_{x_{1}}^{x_{2}}(\int_{-f(x)}^{f(x)}g(t)dt)dx)d\mu$ where $f(x)$ is a convex function defined only in the interval $x_1<x<x_2$ and ...
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1answer
27 views

Convexity of Gaussian Q-function (monotone decreasing)

I have a known convex function. If I take the Gaussian Q-function of this convex function would the resultant function also be convex?
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2answers
51 views

How to get a “Gaussian Ellipse”?

Assume I have a multidimensional Gaussian distribution defined by a mean $\mu$ and covariance matrix $\Sigma$. I want to calculate an iso contour / ellipsoid that is aligned with the shape of the PDF. ...
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18 views

Sufficient Statistic for n-fold multivariate gaussian

I want to show that, for the n-fold product distribution of a d-dimensional gaussian $X_i, i\in \{1,...,n \}$ with mean $\mu \in \mathbb{R}^d$ and covariance matrix $\Lambda \in \mathbb{R}^{d\times d}$...
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17 views

Express the exponent of a multidimensional gaussian as a vector product

I am reading a paper and I am a bit confused by a paragraph in it. The paper introduces early on the formula for multidimensional Gaussians: $$c\cdot e^{-\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu)}$$ ...
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21 views

Joint distribution on norms to p-th power of a Gaussian random vector

If $X$ is a gaussian random vector such that $X \sim \mathcal{N}(0, \sigma^2 I)$, how can you estimate the joint distribution: $$ P(\| X \| < a, \| X \|^2 < b, \| X \|^3 < c), $$ where $a,b,c ...
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1answer
52 views

Commutativity relationship between the Malliavin derivative and the Skorkohod divergence operators.

On proposition $1.3.1$ of Nualart's book "The Malliavin Calculus and Related Topics" it's stated the following. Let $H$ be a real separable Hilbert space, and let $W=\{W(h),h\in H\}$ be a Gaussian ...
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1answer
7 views

Covariance function of a Gaussian Process, sin and cos

Let $X(t) = \varepsilon_1 \cos(t)+ \varepsilon_2 \sin(t)$ where $\varepsilon_1, \varepsilon_2 ∼ N(0, 1)$ iid. Find the covariance function $C_X(s, t)$. I know that E(X) = 0 so Cov(X(s),X(t)) = E(X(s)...
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21 views

Gaussian Distribution for Probability

Say I'm in a store, selling baubles. Each day, 100 people come to the store - each person has a 10% chance of buying a bauble. On average, 10 people will buy a bauble. But is there a gaussian ...
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11 views

Gaussian field indexed by a compact metric space

I found this result in the notes of a course but I can't find any bibliography about it. Could you help me with that? Let $T$ be a compact metric space and $X_{t \in T}$ a centered gaussian field ...
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Can't figure out what this symbol in a Kac-Rice method means

I've been reading a paper and I can't seem to understand what this cursive symbol "N" means at the beginning of the equation below. It's in the context of a staircase function and the Kac-Rice method. ...
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9 views

Mesurable modification

I found this result in a talk about gaussian fields but I can't find any bibliography about it. Could you help me by citing any book or article? Let $(T,\mathcal{T})$ a measurable space and $(X_t)_{t ...
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18 views

How to determine the gaussian distribution of data related to another data

I have a sample of real data with this structure: ANGLE | VALUE 40.2° | 300 41.9° | 350 42.3° | 320 42.9° | 400 The given ...
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1answer
37 views

Integral of $\int_{-\infty}^{\infty} \, \cos^{2}(\theta) f_{\Theta}(\theta) \, d\theta$

What is the value of the integral $$I=\int_{-\infty}^{\infty} \, \cos^{2}(\theta) f_{\Theta}(\theta) \, d\theta$$ where $f_{\Theta}(\theta)$ is the gaussian pdf with mean $0$ and variance $1$
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1answer
34 views

How to understand the covariance matrix in D dimensional gaussian distributions?

When looking at the covariance matrix of a D dimensional gaussian distribution it's intuitively clear that the diagonals have to be equal 1. However when trying to derive the bivariate gaussian for ...
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2answers
80 views

Probability distribution of the infinity norm of a Gaussian vector

Let $N \geq 1$ be an integer. Let $X$ be a standard $\mathbb{R}^N$ Gaussian vector (all components are $\mathcal{N}(0, 1)$ and i. i. d.). Let $A \in \mathcal{M}_N(\mathbb{R})$ be a deterministic ...
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34 views

Independence of norm and direction of Gaussian vectors

Let $x,y\in\mathbb{R}^d$ be i.i.d. $\mathcal{N}(0,I_d)$ distributed, and let $a,b\in\mathbb{R}$ be arbitrary. Then are the random variables $$ \|x\|_2 \quad\text{and}\quad \frac{ax+by}{\|ax+by\|_2} $...
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14 views

Find standard deviation $s$ such that $\mathbb{P}(X \in [-t, t]) = 1 - \alpha$ for $X \sim \mathcal{N}(0, s)$

Let $t >0$ and $\alpha \in (0, 1)$. I am looking for the standard deviation $s > 0$ such that $\mathbb{P}(X \in [-t, t]) = 1 - \alpha$ for any $X \sim \mathcal{N}(0, s)$. Let $s$ be a ...
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2answers
53 views

$\lim_{n \rightarrow \infty} \sum_{k = 0}^n \frac{e^{-n} n^k}{k!}$ =?

By viewing it as the sum of Poisson random variables and using Central Limit Theorem, we might transform the formula to be the probability of a standardized gaussian random variable less or equal to 0....
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2answers
22 views

Generic integral for n-th order gaussian

I'm working with super-gaussian profiles of the form: $$ f(x) = A\exp\bigg({-\big(\frac{(x-x_0)^2}{2\sigma^2}\big)}^n\bigg) $$ I need to integrate this function for various $n$ values (probably ...