Questions tagged [gaussian]

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

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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 ...
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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!
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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 ...
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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{...
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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 ...
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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 $Y$ .
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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 ...
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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 ...
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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 ...
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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 ...
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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'...
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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. ...
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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 $\...
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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 ...
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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|...
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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 ...
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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 ...
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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 ...
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1answer
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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. ...
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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 $$ \...
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1answer
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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?
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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 ...
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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{...
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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 ...
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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}...
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10 views

Moment fucntion of Gaussian process

Was wondering if the recurrence formula for the moment function of a Gaussian process $$ <z(t_1)z(t_2)\cdots z(t_n)> = \sum^n_{m=1}C(t_1, t_m)<z(t_2)\cdots z(t_{m-1})z(t_{m+1})z(t_m)>$$ ...
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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 ...
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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)...
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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 ...
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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}...
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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 ...
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1answer
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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 ...
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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 $\...
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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 ...
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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 ...
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1answer
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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'|...
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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 $m=(1,2)$ ...
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1answer
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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-...
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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}{...
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1answer
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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\...
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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 ...
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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 ...
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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 $\mu(x)$. ...
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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)...
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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.
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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 $$ |e^{-(A+iB)^2}| \leq e^{-A^2} $$
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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) &...
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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 ...
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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_\...
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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&...

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