Questions tagged [gaussian]

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

555 questions
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Looking for a function that approximates a parabola

I have a shape that is defined by a parabola in a certain range, and a horizontal line outside of that range (see red in figure). I am looking for a single differentiable, without absolute values, non-piecewise, and continuous function that can…
10
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1 answer

Evaluating the integral $ \int_0^1 \frac{e^{-y^2(1+v^2)}}{(1+v^2)^n}dv$

I am trying to evaluate the integral $$ \int_0^1 \frac{e^{-y^2(1+v^2)}}{(1+v^2)^n}dv = e^{-y^2}\int_0^1 \frac{e^{-y^2v^2}}{(1+v^2)^n}dv $$ for $n\in \mathbb{N}$.For n=1 one finds Owen's T function, i.e. \begin{align} \int_0^1…
7
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0 answers

An inequality involving two i.i.d. standard Gaussians.

Short question: Let $ r,t,h\ge0 $ and $ G_1,G_2 $ be i.i.d. standard Gaussians. Is it true that $$ \Pr[\{(r+G_1)^2\le t\}\cup\{(r+G_1)^2+G_2^2\le h\}]\le\Pr[\{G_2^2\le t\}\cup\{(r+G_1)^2+G_2^2\le h\}]. $$ Motivation: I formulated the inequality in…
Yihan Zhang
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7
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2 answers

Evaluating $ \int_{-\infty}^{t} e^{-(\tau+a)^2} \mathrm{erf}(\tau) \mathrm{d}\tau$

I need to evaluate this integral: $$ I(t,a) = \int_{-\infty}^{t} e^{-(\tau+a)^2} \mathrm{erf}(\tau) \ \mathrm{d}\tau $$ where the $\mathrm{erf}(\tau)$ is the error function. I can prove that this integral converges. By employing the python…
7
votes
1 answer

Is a Gaussian Processes equivalent to a linear transformation of itself?

I am wondering whether a non-degenerate continuous Gaussian process is equivalent in distribution to a linear transformation of itself. More specifically, let $T$ be a separable, complete and compact metric space, $C(T,\mathbb{R})$ the set of…
6
votes
1 answer

Solution of SDE $dX(t)=a(t)dt+b(t)dW(t)$ is gaussian?

A stochastic process $X(t)$ by definition is gaussian iff all its finite-dimensional joint probability density functions are multivariate gaussian. Namely iff given the times $(t_1,t_2,...,t_n)$ , the random variables $(X(t_1),X(t_2),...,X(t_n))$…
6
votes
1 answer

Fake Brownian motion - not Gaussian

Let $G$ be a standard normal random variable and define two standard Brownian motions $(W_t)_{t \ge 0}$, $\&$ $(B_t)_{t \ge 0}$. Assume $G, (B_t)$ and $(W_t)$ are independent. Moreover, define that process $Y_t$ by $$ Y_t = \begin{cases} B_t, & 0…
6
votes
1 answer

Convolution of tempered distribution($K$) and gaussian. if $K = K*e^{-\pi |x|^2}$, then $K$ is first degree polynomial.

Q : I need to prove that if $K$ is tempered distribution on $\mathbb{R}$ satisfying: \begin{equation} K = K*e^{-\pi |x|^2} \end{equation} then $K$ is first degree polynomial. mean $K(x) = Ax + b$ Remark: The question was changed. The original was…
5
votes
0 answers

Farmer wants to know how wet their field is

Problem A farmer wants a better understanding of rainfall on their field. Assuming rain falls randomly and with equal likelihood over the entire field, the farmer thinks they can model the volume of rainfall $R$ on any given patch as normally…
5
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0 answers

Why should the sum of squares of two Independent normals be memory-less

In section 11.3.1 of Introduction to probability models by Ross (10th edition), a very strange phenomenon is described. If you take two independent standard normal distributions and sum their squares, you get an exponential distribution with rate…
Rohit Pandey
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5
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2 answers

Chi Squared Clarification

Suppose you have $Z_1, Z_2, Z_3$ which are all independent standard Gaussian variables. Suppose you have $$A=\frac{(Z_1-2Z_2+Z_3)^2}{12}+\frac{(Z_1-Z_3)^2}{4}+\frac{(Z_1-Z_2)^2}{4}+\frac{(Z_1+Z_2-2Z_3)^2}{12}.$$ $A$ is $\chi_2^2$, but I don't see…
5
votes
4 answers

Intuitive explanation for $\lim\limits_{n\to\infty}\left(\frac{\left(n!\right)^{2}}{\left(n-x\right)!\left(n+x\right)!}\right)^{n}=e^{-x^2}$

In this post I noticed (at first numerically) that: $$\lim\limits_{n\to\infty}\left(\frac{\left(n!\right)^{2}}{\left(n-x\right)!\left(n+x\right)!}\right)^{n}=e^{-x^2}$$ This can be proved by looking at the Taylor…
tyobrien
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5
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Analyticity of determinant formula for Gaussian integral

It is a well known fact that $\int_{\mathbb{R}^n} e^{-\frac{1}{2}x \cdot A x} dx = \sqrt{\frac{(2\pi)^n}{\det{A}}}$ for real, positive definite $A$. The left hand side of the equation make sense for any complex-symmetric $A$ with real part positive…
4
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1 answer

Comparing the Training Costs of Machine Learning Algorithm: A Mathematical Perspective

Recently, I was looking at the optimization functions required in training Kernel Based Methods compared to Neural Networks. 1) Kernel Methods: For instance, I was looking at the optimization in Support Vector Machines: And Gaussian Process…
4
votes
1 answer

Positive semi-definitness of modified RBF Kernel

Let's say I have an imaging space made of 100x100 pixels and I want to make a covariance matrix using RBF kernel (Gaussian kernel). In other words, say the covariance matrix is $C\in\mathbf R^{10000\times10000}$, then the $i,j$th element is…
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