For questions about the Gaussian probability distribution, its definition, properties and use.
Questions tagged [gaussian]
491 questions
35
votes
7 answers
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…
Gimelist
- 563
10
votes
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…
drandran12
- 472
7
votes
3 answers
Does a Markov chain with Gaussian transitions $p(x_t|x_{t-1})=\mathcal N(\sqrt{1-\beta_t}x_{t-1},\beta_tI)$ tend to $\mathcal N(0,I)$?
The background of this question is a generative process called reverse diffusion process, where one starts with a data distribution $x_0\sim p_{\rm data}(x_0)$ (each sample lies in $\mathbb{R}^D$) and defines a Markov chain (called diffusion…
trisct
- 5,373
7
votes
1 answer
Polar coordinates for a gaussian random vector
I've been finding some prolems in solving exercise 3.3.7 of Vershynin's book "High dimensional probability".
Let $X\sim N(0,I_n)$ standard multivariate normal random vector on $\mathbb{R}^n$ and write it as:
\begin{equation}
X=\|X\|_2…
Pefok
- 684
7
votes
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
- 163
7
votes
2 answers
Tail Probabilities of Multi-Variate Normal
For a standard normal random variable $X \sim \mathcal{N}(0,1)$, we have the simple upper-tail bound of
$$\mathbb{P} (X > x) \leq \frac{1}{x \sqrt{2\pi}} e^{-x^2 / 2}$$
and thus from this we can deduce the general upper-tail bound for $X' \sim…
paulinho
- 6,730
7
votes
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…
Ghoti
- 103
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…
Peter Koepernik
- 744
6
votes
1 answer
A complicated integral that Mathematica can't compute
I have a very complicated distribution function of which I want to find the expected value.
The distribution I got for a function having the cosine of the samples taken from a Gaussian distribution.
$$ y = \cos(x) $$
Where $x$ values are drawn from…
CfourPiO
- 119
- 9
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 $(X(t_1),X(t_2),...,X(t_n))$…
Antonio19932806
- 235
6
votes
1 answer
How does a Gaussian Process define a probability distribution in the functions space?
I am studying Gaussian Process Regression. I will post a text from the book Gaussian Process for Machine Learning, by C. E. Rasmussen & C. K. I. Williams:
We first consider a simple 1-d regression problem, mapping from an input x to an output f (x).…
Fam
- 855
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…
dp1221
- 707
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…
shestak
- 105
5
votes
0 answers
Proof of Multivariate Central Limit Theorem using Cramér-Wold
Even though there are problems here have the same title to my problem but I have different question.
From Jacod-Protter ''Probability Essentials''
Multivariate CLT: Let $(X_j)_{j≥1}$ be i.i.d. $,\mathbb{R}^d$-valued r.v. Let the (vector) $\mu =…
JEAD MACALISANG
- 139
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…
Greedo
- 201