Questions tagged [gaussian]
For questions about the Gaussian probability distribution, its definition, properties and use.
352
questions
0
votes
0answers
9 views
Expression for Pareto Frontier with Selection from Joint Normal
Suppose you're admitting a fixed fraction of some population using a linear function of their characteristics, can you characterize the Pareto frontier from the correlation of characteristics?
Each ...
0
votes
2answers
13 views
Upper bound on difference between Gaussian CDFs
May I know if there are any non-trivial upper bounds $f$ on the following:
$$\Phi(a + \Delta) - \Phi(a) \leq f(a, \Delta)$$
for $\Phi$ the CDF of a standard normal and all $a, \Delta > 0$.
Thanks!
0
votes
1answer
17 views
How is the linear combination of Normal Distribution also normal?
X ~ N(μ, σ2)
z-square = (X-μ)/σ ~ N (0,1)
If Y = (X - α) / β
X = α + βY
Y ~ N((μ - α) / β, (1 / β^2) * σ2)
How can we derive from above two that linear combination of normal distribution is also ...
0
votes
0answers
17 views
Compute Convariance Function for a given model $f(x)=x^a$
Consider a model
\begin{equation}
\mathbf{X} = x^a,\, \text{where} \,\,\, a \sim N(1,w).
\end{equation}
It's mean is
\begin{equation}
E[\mathbf{X}] = x \exp\left\{\frac{(w\log x)^2}{2}\right\}.
\end{...
6
votes
1answer
70 views
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 ...
-1
votes
1answer
33 views
Find probability density function given Gaussian random variable [closed]
A Gaussian random variable $X$ has mean $2$ and variance $16$, which we often write as $X \sim N(2, 16)$.
Let $Y = 3X + 5$. How do I find $fY (y)$, the probability density function of $Y$ .
0
votes
0answers
8 views
Deriving estimator for non gaussian linear model
Gaussian linear models are often insufficient in practical applications, where noise can be heavy- tailed. In this problem, we consider a linear model of the form yi = a · xi + b + ei. The (ei) are ...
0
votes
0answers
13 views
Gaussian Kernel Expectation
Let $\kappa(x, x')=\exp(-\frac{1}{2\sigma^2}||x-x'||)$ be the Gaussian kernel with the multivariate random variable $x, x' \in \mathbb{R}^D$ and scalar parameter $\sigma \in \mathbb{R}_+$. I am ...
2
votes
1answer
27 views
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 ...
0
votes
0answers
13 views
isoelastic expectation of normal random variable (multivariate)
I have the following random variable which is multivariate normally distribued with mean zero and variance $\Sigma$:
$\epsilon \sim N(0,\Sigma)$. Its dimension is $K$x$1$.
I need to calculate the ...
0
votes
1answer
41 views
Why is the derivative of this Gaussian Function negative before being positive?
From pg. 61 of Principles of Quantum Mechanics, the Guassian $g_Δ$ is defined as
$$
g_Δ(x-x') = \frac{1}{(πΔ^2)^{1/2}} \exp \left[ -\frac{(x-x')^2}{Δ^2} \right]
$$
where the Gaussian is centered at $x'...
0
votes
1answer
24 views
Prove that a Truncated Gaussian is log-concave
Is the truncated Gaussian a log-concave distribution?
I read here at Section 2.2.5 that whatever log-concave truncated distribution (and hence also Gaussian) is log-concave and the proof is trivial. ...
1
vote
0answers
43 views
Gaussian Mean Testing
Consider
\begin{align}
\mathbb{P} \left( ||Y_n||_2 \leq \frac{\epsilon}{2}\right) \tag1
\end{align}
where $Y_n \in \mathbb{R}^d$ is Gaussian with mean $\mu \in \mathbb{R}^d$ and covariance matrix $\...
0
votes
0answers
72 views
+100
Understanding algorithm to show $(x+\partial_x)e^{-x^2/2}=0$ using python?
I am given the following Python code which is supposed to verify numerically that
$$(\partial_x + x)e^{-x^2/2}=0.$$
The algorithm does this, by transforming everything into a Fourier basis and then ...
-1
votes
1answer
50 views
Find posterior distribution and find marginal distribution
We have
z = H s + $\epsilon$
with $\epsilon \sim N(0, R)$ and $s \sim N(\mu, Q)$ where R and Q are symmetric and positive definite. I'm trying to derive the posterior distribution. I have p(s) and p(z|...
0
votes
0answers
39 views
Gaussian with poles (complex contour integration)
I am considering the integral
$$ I = \int_{-\infty}^\infty \text{d} x \: \frac{f(x)}{x}e^{-x^2}, $$
where $f(x)$ is an even function of $x$ with no singularities/poles. Due to the (anti)symmetry of ...
2
votes
0answers
32 views
How did Gauss come up with this function?
I have been studying statistics lately, and after having my first brush with normal distributions, I was curious to know more about it. I have researched about the history of normal distributions and ...
1
vote
0answers
36 views
Gaussian deconvolution for rapidly decreasing functions.
Gaussian convolution with variance $v$ is defined as
$$
{\cal G}_v[f](x):=\int_{-\infty}^{\infty}\frac{1}{\sqrt{2\pi v}}f(y) e^{-\frac{(y-x)^2}{2v}}dx.
$$
Given a function $g$, does there exist a a ...
4
votes
1answer
87 views
Does this weighted sum of Gaussians converge to $1/x^2$?
By coincidence I discovered that a weighted sum of several Gaussians seems to converge to $1/x^2$. Now I wonder whether that's a well-known property, just a coincidence or if this could be proven.
...
0
votes
0answers
14 views
Correlation matrix of Gaussian copula
The question is related to the Gaussian copula.
Let $\Phi(x)$ denote the cdf of standard Normal distribution. Let $(X_1, X_2) \sim \mathcal{N}(0,\Sigma)$ be joint Normal with covariance matrix
$$
\...
1
vote
1answer
18 views
How can you prove that a conditional bivariate Gaussian is a univariate Gaussian?
Is there a way to prove that a bivariate Gaussian becomes a univariate Gaussian when conditioned on one of the two variables?
0
votes
0answers
38 views
Telling a single from a double Gaussian model
I want to implement a statistical test to tell if a small sample of a univariate data (say $25$ values) is drawn from a Gaussian distribution or from a mixture of two Gaussians. None of the parameters ...
3
votes
0answers
30 views
Gaussian with prior $p(\mu, \Sigma) \propto |\Sigma|^{-(d+1)/2}$. Why is the posterior $\Sigma \mid y \sim \text{Inv-Wishart}(S^{-1})$?
According to Gelman et al book, page 73, if
$p(\mu, \Sigma) \propto |\Sigma|^{-(d+1)/2}$
then
$\Sigma \mid y \sim \text{Inv-Wishart}(S^{-1})$ with $n-1$ degrees of freedom
for
$$S = \sum_i (y_i - \bar{...
1
vote
0answers
24 views
covariance matrix with parameters to guarantee independency?
Suppose we have two gaussian vectors $W$ and $Z$ such that their covariance matrix is
$C_{W,Z}$=$$\begin{bmatrix}
4 & \alpha-1 \\
\alpha-1 & 1
\\\end{bmatrix}
$$
the question ...
1
vote
0answers
7 views
Random Variable with Sub-gaussian law - How to estimate density
I have n random variables $X_0^1,...X_0^n$ which are i.i.d. with law $\mu_0$ such that
\begin{align*}
\exists\;\epsilon>0:\; \mu_0(\lambda,\infty)=\mathcal{O}(\exp\{-\epsilon\lambda\})\quad\text{as}...
0
votes
0answers
10 views
Moment fucntion of Gaussian process
Was wondering if the recurrence formula for the moment function of a Gaussian process
$$ <z(t_1)z(t_2)\cdots z(t_n)> = \sum^n_{m=1}C(t_1, t_m)<z(t_2)\cdots z(t_{m-1})z(t_{m+1})z(t_m)>$$
...
0
votes
0answers
16 views
Gaussian function with dependent variables
Through my studies so far, I was able to understand the Gaussian function when variables are independent in a normal distribution, but I cannot think of cases where they are not independent.
I would ...
2
votes
0answers
21 views
Operator on Sobolev Space w.r.t. Gaussian measure
Let $X=H_r^p(\mathbb{R},\mu)$ be the Sobolev space of order $r$ with respect to the Gaussian measure $\mu$.
Is the operator $d\colon H_1^p(\mathbb{R},\mu)\to L^p(\mathbb{R},\mu)$ given by $df(x)=-f'(x)...
0
votes
0answers
12 views
Gaussian Process averaging
I am currently working on a regression problem and I want to use gaussian processes. Unfortunately my dataset is too large so my plan was to split the data and train several GPs. In the end I want to ...
0
votes
0answers
39 views
Average probability of detection error with Gaussian noise vector
I have a problem with how to express the average probability of error when The Probability involves vectors. The problem is a classic detection problem where the probability under the Hypothesis $S_{k}...
2
votes
1answer
45 views
Brownian motion exercise
Given $Z(t)=B(t)Y$, with $B(t)$ standard brownian motion, also $t \in (0,1)$ and $Y \sim N(0,1)$. $B(t)$ and $Y$ are independent.
Compute $E[Z(t)]$ , $Var[Z(t)]$ and $Cov[Z(t),Z(s)]$ .
Prove also that ...
1
vote
1answer
17 views
How do you deal with units when normalizing a Gaussian?
I have a Gaussian with unitless output, but which is a function of time. The formula for this function is
$$
f(t)=\exp{(-\frac{t^2}{2c^2})}
$$
Where both $t$ and $c$ are in units of time. I want to ...
1
vote
0answers
10 views
Norm of regularized least square estimator
Let $\theta_*$ be the d-dimensional hidden true parameter,
$y_t = x_t^\top \theta_* + \eta_t$ where $\eta_t$ is a standard gaussian noise.
It is well known that regularized least square estimator is $\...
1
vote
0answers
48 views
What is the “t” value in a Gaussian CDF, and how do I calculate it?
I was reading up on the formula for a Gaussian CDF:
$\Phi(x) = \frac12 [1 + $erf$(\frac{x}{\sqrt{2}})]$
where erf(x) is defined as:
erf$(x) = \frac{2}{\sqrt{\pi}} $$\int_0^x e^{-t^2} \,dt$
My question ...
1
vote
1answer
41 views
Proving two random vectors are following the same distribution using characteristic function?
Suppose we have two centered Gaussian random vectors $X$ and $Y$ I.I.D.
Find the distribution of $X_a=X\cos(a)+Y\sin(a)$ and $Y_a=-X\sin(a) + Y\cos(a)$ with $a$ being in $[0,2\pi)$.
What I tried to do ...
2
votes
1answer
75 views
How to estimate E$[\alpha-\alpha']$ and Var$[\alpha-\alpha']$ conditioned on $\alpha>\alpha'>0$ where $\alpha$ and $\alpha'$ are iid standard normal?
Let $\alpha$ and $\alpha'$ be independent standard normal random variables, how to estimate the conditional mean $\mathbb{E}[\alpha-\alpha'|\alpha > \alpha' >0]$ and variance Var$[\alpha-\alpha'|...
1
vote
1answer
29 views
Finding the covariance matrix of two random gaussian vectors and their characteristic function
Let $(W,Z)$ be the gaussian random variable vector that we want to find its covariance matrice and characteristic function.
We have $(X,Y)$ a guassian random variable vector with a mean of $m=(1,2)$ ...
2
votes
1answer
19 views
Spacing between largest eigenvalues in the GUE
Consider GUE$(N)$, the set of $N \times N$ random Hermitian matrices where the elements on the diagonal are i.i.d. according to $\mathcal{N}(0,1)$ and both the real and imaginary parts of the upper-...
1
vote
0answers
13 views
If $G^{(k)}(y)$ is the $k^{th}$ derivative of the Gaussian function, how can we guarantee $|G^{(k)}| \leq C_k e^{-|y|}$ for some coefficients $C_k$?
I came across the inequality in the question in Fourier Analysis and It's Applications by Folland, specifically in the section on Convolutions. If the Gaussian function is defined as
$$G(y) = \frac{1}{...
2
votes
1answer
40 views
How to prove geometrically that anticorrelated Gaussians are separated?
I am trying to prove the following theorem:
There exists a universal constant $K>0$ such that if $g,h$ are standard Gaussians with $\mathsf{E} gh=1-\alpha$,
$$\mathsf{P}\left(g< -1,h> 1\...
0
votes
0answers
19 views
How to treat a Log-Normal distribution as Normal distribution?
I have some data X that is log-normally distributed. I also have a Bayesian process that computes the posterior (normal if normal prior). Can I fit a normal distribution to X's log, analytically ...
0
votes
1answer
26 views
Converting a daily standard deviation to an annual one
This post discusses how to convert a daily standard deviation into an annualized one.
Standard Deviation Annualized
However, it doesn't include a proof.
I've tried to prove it myself by using a ...
1
vote
0answers
33 views
What is the gap between average values in a Gaussian Process and the real mean values?
Let's consider a function $f(x)$ where $x \in \mathcal X$, and $f(x)$ is sampled from a Gaussian Process. Also assume that for each $x\in\mathcal X$, $f(x)$ has a fixed unknown mean which is $\mu(x)$.
...
0
votes
0answers
16 views
Sobolev (Gaussian) regularity for the solution of this PDE.
I am dealing with the following pde (for simplicity I am considering here just the unidimensional case)
$$\partial_t u+\left(\frac{\sigma(t)}{a}\right)\partial_x u= b(t,u)+ \sigma(t) u\cdot x,\; u(0,x)...
1
vote
0answers
39 views
Standard Gaussian Distribution
What would P (X=Y) be in a joint standard gaussian distribution?
I was thinking to integrate however I am not sure what values to use.
0
votes
1answer
62 views
Modulus of $\exp(-(A+iB)^2)$
I know modulus of $e^z$ is just $e^{\Re(z)}$, and that $|A+iB| \geq A^2$. What I want to know is if the following holds true
$$
|e^{-(A+iB)^2}| \leq e^{-A^2}
$$
0
votes
1answer
14 views
Expectation of Cumulative Gaussian Function
If M(x) is the cumulative Gaussian function and X is N(0,1) then what is E[M(X)]?
Thus if X ~ N(0,1):
$M(x) = P(X \leq x)=\Phi(x)$
The answer given is for x in (0,1):
\begin{align} P(M(X) \leq x) &...
0
votes
1answer
27 views
Proof of joint Gaussian distribution in a linear equation
Let $X = AY + Z$ where $A$ is constant, $Y$ and $Z$ are independent gaussian rvs. How do you prove that $X$ and $Y$ are jointly gaussian? I know that the sum of two independent gaussians gives a ...
1
vote
1answer
37 views
Prove that Cameron–Martin space is set of elements with finite Cameron–Martin norm
Let $\mu$ be a centered Gaussian measure on a separable Banach space $\mathscr{U}$. The construction of the Cameron-Martin Hilbert space $H_\mu \subset \mathscr{U}$ typically goes as follows: Let $C_\...
-5
votes
1answer
61 views
Can $\frac{1}{1+x^2}$ ever become a gaussian? [closed]
I guess this is equivalent to asking if the $\frac{2}{\pi}\arctan x$ can ever become $\operatorname{erf}(x)$ ?
$\frac{2}{\pi}\arctan 2x$ seems to be rather close, but the tails are still too "big&...
