All Questions
Tagged with pr.probability gaussian
220 questions
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152
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The monotonicity of the bivariate normal with non-isotropic covariance
Let $Y = (Y_1, Y_2) \sim N(0, 11^T + I)$, be a bivariate normal random variable with non-isotropic covariance.
Define $y = (y_1, y_2)$ and let
\begin{align}
F_{\delta}(y) = \Pr[Y_1 > y_1 - \delta, ...
- 113
1
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1
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185
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Cameron-Martin space of product space
Suppose you have Banach spaces $\mathcal B_\alpha$ where $\alpha$ is in some index set $I$. Let $\mu_\alpha$ be Gaussian measures on $\mathcal B_\alpha$ with Cameron-Martin spaces $\mathcal H_{\mu_\...
- 1,904
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1
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161
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Conditional Gaussians in infinite dimensions
I asked this over on cross validated, but thought it might also get an answer here:
The law of the conditional Gaussian distribution (the mean and covariance) are frequently mentioned to extend to the ...
- 379
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1
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613
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Integral of the product of a gaussian pdf and cdf
I am trying to solve the integral of a gaussian cumulative distribution function and a gaussian probability function. On this site I have seen solutions of similar, less general integrals (e.g. ...
- 11
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1
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101
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Estimating the average of two gaussians' mean with minimal squared error
This is a follow-up to my previous question.
Assume that $X\sim \mathcal N(\mu_1,\sigma_1^2)$ and $Y\sim \mathcal N(\mu_2,\sigma_2^2)$.
I want to estimate $\frac{\mu_1+\mu_2}{2}$ after observing $X,Y$....
- 618
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1
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666
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Definite integral of 2d Gaussian
Is there some analytic expression or even an approximation of the definite 2D Gaussian integral of the form: $$E=\int_a^b Dg \int_{cg+d}^\infty Dh$$ where $Dg=\frac{dg}{\sqrt{2 \pi}} e^{-g^2/2}$ and a,...
- 373
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1
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169
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Probability involving dependent random variables constructed from i.i.d. Gaussians
This is a problem I need to address for a certain computation in my research.
Let $Y_1,\dots,Y_n$ be a sequence of i.i.d. standard normal variables; and let $I\subset[0,+\infty)$ be an interval. In my ...
- 1,096
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1
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287
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Random matrix and spherical spin-glass
The Hamiltonian of the p-spherical spin glass model is
$$H_{N,p}(\sigma)=\frac{1}{N^{\frac{p-1}{2}}} \sum_{i_1,...,i_p=1}^N X_{i_1,...,i_p} \sigma_{i_1}\cdot...\cdot \sigma_{i_p}$$
where $\sigma \in ...
- 536
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1
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115
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Supremum of centered jointly generalized chi-square random variables
Let $\zeta_n$ be a sequence of centered jointly generalized chi-square random variables, i.e. $\zeta_n = \sum_{k=1}^{m_n} a_{k,n}(\xi_{k,n}^2 - E[\xi_{k,n}^2])$, and $\xi_{k,n}$ are centered jointly ...
- 1,533
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1
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207
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Computing probability that $Ax\geq0$ where $x$ is a vector of iid gaussians and $A$ is matrix of $1$s and $0$s
This question came up in my research: What is the probability that $Ax\geq0$ where $x$ is a vector of iid gaussians and $A$ is matrix of $1$s and $0$s?
So far I only figured out that I can do Monte ...
- 113
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0
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56
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Quantitative multivariate CLT from quantitative CLT of linear combinations
Suppose $Z_1, \ldots, Z_k$ are random variables with mean $0$ and variance $1$ that are "approximately jointly Gaussian" in the sense that for any scalars $c_1, \ldots, c_k$, we have that $\...
- 111
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59
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Gaussian Hypercontractivity of Chaos based on Gaussian with value in Hilbert spaces?
The classical Gaussian hypercontractivity is stated as following: Suppose $\xi$ is a Gaussian variable and $H_n(\xi)$ is the space of n-th homogeneous Wiener chaos constructed from $\xi$, then for any ...
- 253
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133
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A question about one Malliavin derivative calculation
Recently, I've asked here a question. While trying to find an answer on my own, I found an idea which I now will briefly describe below. I am not familiar enough with the Malliavin calculus, so my ...
- 620
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Deterministic multifractal measure with quadratic singular spectrum?
For a non-negative locally finite measure $\mu$ on a bounded metric space $(\Omega,\mathcal{B})$, its local Holder exponent $f(x)$ is defined as $$f(x)=\lim_{r\downarrow 0}\frac{\mu(B(x,r))}{\log r}$$
...
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241
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Expectation of top-K selection of squared Gaussian random variables
Let us have
$$
Z = [z_1, z_2, \dots, z_n],
$$ where $z_i \sim N(0, \sigma^2)$ and are iid. Additionally, consider
$$
X_k := \{ x \in \{0, 1\}^n : e^T x = k \}
$$ If $Y = \max_{X \in X_k} |Z^T X|^2,$ ...
1
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1
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180
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Conditional differential entropy of sum of Gaussians
Is it possible to give an expression for the conditional differential entropy $h(A+B\mid C+D),$ where $A,B,C,D$ are normally distributed with known standard deviations $σ_A,\ldots,σ_D$ and where all ...
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0
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168
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Expectation of inverse of complex Gaussian variables
If we consider a complex Gaussian random variable as $h\sim\mathcal{CN}(0,\gamma)$, where $\gamma$ is the variance. Is there any closed-form solution with $\gamma$ for $\mathbf{E}\left[\frac{1}{\lVert ...
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100
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Ito formula for fractional BM + drift and supremum bound
Let $W^H$ be a fBm with Hurst parameter $H$ and let $\mathcal{H}$ be its Cameron-Martin space. Then by Girsanov theorem we know that if $\mathbb{P}$ is an fBm measure, it holds that there exists a ...
- 111
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43
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Generalization of a Gaussian measure continuity result from Hilbert to Banach space
Da Prato/Zabczyk "Second Order Partial Differential Equations in Hilbert Spaces" states the following lemma (this is a reformulation of proposition 1.3.11 in their book):
Let $\mu = \mathcal ...
- 375
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1
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417
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Obtaining the error term of binomial distribution's entropy from the differential entropy of a Gaussian distribution
It is known that the first order error term in the Shannon entropy formula for a binomial distribution is $1/n$ (for example, see the Wikipedia page Binomial distribution), where in the limit $n \to \...
user325283
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176
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Gaussian order statistics
Setup. Let $\alpha\in(0,1)$ fixed; and $\tau\in[0,1]$ (think of it very close to one).
Suppose $X_1,\dots,X_n$ are i.i.d. standard normal.
Let $Y_1,\dots,Y_n$ be another sequence of standard normals ...
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0
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121
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Relation satisfied by a Gaussian random variable
I want to prove the following relation for $X\sim \mathcal{N}(0,1)$, $x\in \mathbb{R}$ and $f(x)=\mathbb{E}[\max(X,x)]$:
$$f(\frac{f(x+1)+f(x-1)}{2})\leq \frac{f(f(x)-1)+f(f(x)+1)}{2}$$
It seems that ...
- 171
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1
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66
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Comparing noisy truncated RV with noisy regular RV
For some reason, I'm having difficulties proving something that is intuitively simple.
Assuming I have two a random variable, $x$ and $x^{truncated}$, where $x^{truncated}$ is the truncated version of ...
- 183
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0
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62
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Distances between up and down crosses in Gaussian Processes
Given a gaussian process $g := \mathcal{GP}\left(\mu, \Sigma \right)$,
where $\mu$ is the mean and $\Sigma$ is the covariance function, I am interested in estimating the mean value $L_m$ of the ...
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79
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Showing that additive Gaussian noise never increases sparsity
Let $\mathbf{1}\in\mathbb{R}^d$ be the $d$-dimensional all-ones vector and let $n\sim\mathcal{N}(0, \sigma^2 I_{d\times d})$, show that
$$ \frac{\| \mathbf{1} + n \|_1}{\|\mathbf{1} + n \|_2} \ge c \...
- 392
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100
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Lower bound for the probability that a certain component of a Gaussian vector dominates all others
Let $X\sim\cal N(\mu,\Sigma)$ be an $n$-dimensional Gaussian vector. I would like to estimate $$P(X_1>\max_{k=2,\dots,n}X_k).$$
While no closed form solution exists (see e.g. MO question on ...
- 959
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0
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104
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Efficient evaluation of multidimensional kernel density estimate
Edit I have copied this discussion to the stats community site here, since I feel it is more relevant. Please feel free to close this in due course.
I've seen a reasonable amount of literature about ...
- 161
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116
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How to compute the following probability involving 4 normal random variables?
$\alpha, \alpha', \beta$ and $\beta'$ are four independent standard normal random variables, I am wondering how to compute the probability of the following two events:
$\alpha>\alpha'>0, \ \ \...
- 327
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2
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135
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Expectation of supremum of sub gaussians
I am trying to prove Lemma 2.3 of ON THE SPECTRAL NORM OF
GAUSSIAN RANDOM MATRICES, which states that
Let $X_1,\cdots,X_n$ be not necessarily independent random variables with $\mathbb{P}[X_i > x] ...
- 150
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1
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806
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Concentration of $\ell_2$ norm of a vector sampled from a distribution
Let $X=(X_1,\ldots,X_n)$, where $X_i \sim P_{p_i}(0,\frac{1}{\lambda})$ are iid, $P_{p_i}$ is sub gaussian distribution for $i^\text{th}$ element, and 0 and $1/\lambda$ are mean and variance.
I'm ...
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194
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Gaussian integral $\int_X \|x\|_X^2 \mu(dx)$ in Banach space
For a centered Gaussian measure $\mu$ on a Hilbert space $X$, it is known that
$$\int_X \|x\|^2 \mu(dx) = tr(Q)$$ where $Q$ is the covariance operator. Is there a similar version for Gaussian measures ...
- 375
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1
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209
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Distribution of the direction of Gaussian random variable
Let $X$ be a complex normal random variable. (Or, equivalently, a 2D real normal.) Is it possible to say anything useful about the distribution of the phase of $X$? Is it possible to do estimation on ...
- 1,069
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1
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69
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Correlation for a Sum of random vectors from the sphere multiplied by matrices
Let $A_1,\dots,A_n\in \mathbb{R}^{d\times d}$ be some matrices. Suppose we sample $x_1,\dots,x_n,y\sim \mathcal{U}(\mathbb{S}^{d-1})$, where $\mathcal{U}(\mathbb{S}^{d-1})$ is the uniform distribution ...
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2
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239
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Computing the expectation of a quadratic matrix form involving Bernoulli and Gaussian distributed matrices
I am working with two random matrices, $Z$ and $H$:
$Z$ is an $n \times K$ matrix with entries sampled i.i.d. from a Bernoulli distribution: $Z_{ij} \sim \mathrm{Bernoulli}(p)$.
$H$ is a $K \times K$ ...
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87
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Is the $2$-point function translation invariant for general Gaussian meaures?
Let us consider the real Hilbert space $H:=L^2\bigl(\mathbb{R}^n, \mathbb{R}^n\bigr)$ and "any" centered Gaussian measure $d\mu$ on it.
Next, denote a generic element of $H$ by the column ...
- 3,477
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51
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Convergence of Gaussian measures $\{ d\mu_a \}$ whose variances depend smoothly on the index $a$
Let $f: \mathbb{R} \to \mathbb{R}$ be a smooth function such that $f(x)$ is positive in a small punctured neighborhood of $x=0$ but $f(0)=0$.
Now, define a collection of centered Gaussian measures on $...
- 3,477
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1
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110
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Positivity of linear combination of gaussian variables
Consider a collection of independent standard Gaussian variables $w_i$ for $i = 1, 2, \ldots, N$. Define its linear combination $f:=\sum_{i=1}^Na_iw_i+b_i$, where $a_i=pb_i$ ($p$ is a fixed parameter),...
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2
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874
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Bounds for the sum of dependent gaussian random variables
Let $X_1,...,X_n$ be $n$ gaussian random variables $N(0,1)$ not necessarily independent or jointly correlated, $S=\sum_{i=1}^n w_i X_i$ be the weighted sum of these gaussian variables (because $(X_i)_{...
- 250
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100
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Expressing a multivariate normal distribution as a mixture of uniform distributions?
Context: Given a scalar normal distribution $X\sim \mathrm{N}(\mu, \sigma^2)$, it is possible to express $X$ as a mixture of uniform distributions over intervals (compound probability distributions), ...
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1
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114
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Ball in separable Banach space has positive Gaussian measure
I have (presumably non-degenerate) Gaussian $\mu$ over separable Banach space $X$. I would like to prove that for any ball of radius $r$ centered at $x$, $\mu(B_r(x))$. I know how to prove this in ...
- 379
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1
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126
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Stationary distribution of AR(1) processes and Lyapunov central limit theorem
Let $X_t$ follow the following AR(1) process:
$$
X_t=\rho X_{t-1}+e_t
$$
in which $|\rho|<1$ and $e_t$ is iid noise term with density $f$, mean $0$ and finite moments up to a certain order.
I am ...
- 157
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1
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102
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Lower bounds for truncated moments of Gaussian measures on Hilbert space
Let $\mu_C$ be a centered Gaussian probability Borel measure on a real separable Hilbert space $\mathcal{H}$ with covariance operator $C$. Denote the ball with radius $r$ in $\mathcal{H}$ centered at ...
- 505
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1
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85
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Conditioned on the expectation and covariance, is the total variation distance maximal for Gaussian distributions?
I want to find two distributions $p_1, p_2$, whose total variation distance is the largest between all pairs of distributions whose expectations $\mu_1, \mu_2\in \mathbb{R}^d$ and covariances $\...
- 265
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1
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61
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What can we say about the order of convergence of a critical point of Gaussian mixture density to its limit when the parameter $h$ goes to $0?$
Density of Gaussian mixture with $n$ components is given by:
$$f(x):=C \sum_{i=1}^{n}e^{-\frac{1}{2}||\frac{x-x_i}{h}||^2}, x_i \in \mathbb{R}^d, h > 0$$
where $C$ is a normalization constant ...
- 1,467
0
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1
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115
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Order of orthant probabilities in a prolate multinormal distribution
This is inspired by the negative answer to the conjecture in Which orthant probabilities are the largest? (For a multivariate normal distribution).
Suppose $X$ has the $k$-dimensional multivariate ...
- 4,219
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1
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204
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Given correlated Gaussian random variables, how to bound the probability that the first is the largest?
Let $Z\sim \mathcal{N}(\mu, \Sigma)$ be a Gaussian random vector in $\mathbb{R}^d$.
What are some nontrivial bounds on
$p=\mathbb{P}(S)$, where $S$ is the event $Z_1=\max_i Z_i$?
This is motivated by ...
0
votes
1
answer
73
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Algorithm for economically sampling method for Gaussian matrix product
Let $A$ be an $n\times n$ random matrix with i.i.d. $N(0,\sigma)$ entries, for some $\sigma>0$ and let $x\in \mathbb{R}^n$. A direct computation shows that $Ax \sim N(0,\sigma x^{\top}x)$.
I would ...
- 5,405
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1
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102
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Sign of expectation value
Consider a multivariate Gaussian-type measure $$d\lambda(x):=\nu_{\mu,\Sigma} e^{-\langle (x-\mu), \Sigma^{-1}(x-\mu) \rangle - \vert x \vert^2} $$
with vector $\mu \in \mathbb R^n$ and $\Sigma$ ...
- 536
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1
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151
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Can an unskewed distribution be expressed as product of a normal and another distribution?
Let $x$ be a continuous random variable with zero mean and zero skew. What are the conditions under which we can say that $x$ can be expressed as the product $z y$ where $z$ is a standard normal and $...
- 620
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1
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752
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transform a polynomial into another one upto a constant
I have a polynomial $p(x)=a_Nx^N+a_{N-1}x^{N-1}+\dots+a_0$. I want to convert this into another polynomial of same order, say $b_Ny^N+b_{N-1}y^{N-1}+\dots+b_0$. Is it possible to find a transformation ...
- 169