# Questions tagged [gaussian]

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

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 ...