All Questions
Tagged with pr.probability gaussian
220 questions
1
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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
2
votes
1
answer
872
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Estimating the average of two gaussians' mean
Assume that $X\sim \mathcal N(\sigma_1,\mu_1)$ and $Y\sim \mathcal N(\sigma_2,\mu_2)$.
I want to estimate $\frac{\mu_1+\mu_2}{2}$ after observing $X,Y$.
In my setting, $\sigma_1,\sigma_2$ are known ...
- 128k
1
vote
0
answers
43
views
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
4
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1
answer
771
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Maximal component of a multivariate Gaussian distribution
Suppose you have a general random Gaussian vector $\mathbf{X}\sim\mathcal{N}\left(\boldsymbol{\mu},\boldsymbol{\Sigma}\right)$. I'm looking for the simple way to calculate the distribution of the ...
- 15
6
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1
answer
237
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Ordering preference for two zero mean Gaussian outcomes
Let $X\sim \mathcal{N}(0,1)$ be a standard Gaussian random variable. If we let $f_a(x)\triangleq\mathbb{E}[\max\{aX,x\}]$ for $a,x >0$, how to prove that $$f_a(f_b(1))<f_b(f_a(1))~~\text{for }0&...
- 61.9k
1
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1
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178
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Tail bound on the RKHS norm of a zero-mean Gaussian process
Let $f \sim \mathcal{GP}(0, K)$ be a zero-mean Gaussian process defined on a compact set $\mathcal{D} \subset \mathbb{R}^d$, where $K \colon \mathcal{D} \times \mathcal{D} \rightarrow \mathbb{R} $ is ...
- 128k
2
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1
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124
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Limiting behavior of $k^{th}$ order statistics of n non-i.i.d chi square random variables
This is related to one of my previous questions here.
Let $(Z_1, Z_2, \ldots, Z_n)\sim N(0, \Omega)$, where $\Omega = (1-\mu) I_{n\times n} + \mu \boldsymbol{1}_n\boldsymbol{1}_n^\top $. Here $\...
- 574
1
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1
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417
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Obtaining the error term of binomial distribution's entropy from the differential entropy of a Gaussian distribution
It is known that the first order error term in the Shannon entropy formula for a binomial distribution is $1/n$ (for example, see the Wikipedia page Binomial distribution), where in the limit $n \to \...
CommunityBot
- 1
2
votes
1
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256
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About a mixture
Consider the following mixture model for a univariate density function
$$
(1) \quad f(x)=\int_{(m, \sigma^2)\in D} g(x; m, \sigma^2) \mu(d(m, \sigma^2))
$$
where $D$ is a compact subset of $\mathbb{R}\...
- 108
2
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3
answers
999
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Sum of Square of the Eigenvalues of Wishart Matrix
Let $A\in\mathbb{R}^{m\times d}$ matrix with iid standard normal entries, and $m\geqslant d$, and define $S=A^T A$.
I want to have a tight upper bound for $\sum_{k=1}^d \lambda_k^2$, where $\...
- 7,514
1
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0
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176
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Gaussian order statistics
Setup. Let $\alpha\in(0,1)$ fixed; and $\tau\in[0,1]$ (think of it very close to one).
Suppose $X_1,\dots,X_n$ are i.i.d. standard normal.
Let $Y_1,\dots,Y_n$ be another sequence of standard normals ...
- 63.9k
1
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1
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666
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Definite integral of 2d Gaussian
Is there some analytic expression or even an approximation of the definite 2D Gaussian integral of the form: $$E=\int_a^b Dg \int_{cg+d}^\infty Dh$$ where $Dg=\frac{dg}{\sqrt{2 \pi}} e^{-g^2/2}$ and a,...
CommunityBot
- 1
5
votes
1
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225
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Anti-concentration of Gaussian when conditioning on event
Let $v$ be a given vector with $\|v\|_{\Sigma^{-1}} \leq 1$, where $\Sigma$ is a positive semi-definite matrix and $\|v\|_{\Sigma^{-1}} = \sqrt{v^\top\Sigma v}$. Meanwhile, let $u$ be a random vector ...
- 128k
4
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3
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345
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Do subgaussian variables obey the slightly-stronger-than-Chernoff tail bound?
If $X \sim Normal(0,1)$, then we have the tail bound:
$$ (*) \qquad\Pr[X > t] \leq \mathcal{O}\left(\frac{e^{-t^2/2}}{t}\right) .$$
Now for general variables $X$, a nice condition is that $X$ be ...
- 128k
1
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0
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121
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Relation satisfied by a Gaussian random variable
I want to prove the following relation for $X\sim \mathcal{N}(0,1)$, $x\in \mathbb{R}$ and $f(x)=\mathbb{E}[\max(X,x)]$:
$$f(\frac{f(x+1)+f(x-1)}{2})\leq \frac{f(f(x)-1)+f(f(x)+1)}{2}$$
It seems that ...
- 171
2
votes
1
answer
1k
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measure of a degenerate Gaussian distribution
I want to do computations with a degenerate Gaussian measure, but I do not know how to represent it
in a close form.
After starting with a Gaussian random variable and restricting it to a condition, I ...
- 128k
3
votes
1
answer
139
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Design a random variable which has the maximal correlation with another random variable
$Y$ is a Gaussian distributed random variable with zero mean and known variance: $Y\sim N(0,\sigma_y)$. We measure $Y$ with a sensor, which is corrupted by white Gaussian noise: $Z=Y+V$; $V\sim N(0,\...
CommunityBot
- 1
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votes
0
answers
86
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A non trivial example of a Gaussian semi-Markov process?
Let $(\Omega, \mathcal A, \mathbb P)$ be a probability space and $X=(X_t)$ a real Gaussian stochastic process.
Let $\mathcal F=(\mathcal F_t)$ be the filtration generated by $(X_t)$.
$X$ is Markov ...
- 66.3k
23
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7
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5k
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What makes Gaussian distributions special?
I'm looking for as many different arguments or derivations as possible that support the informal claim that Gaussian/Normal distributions are "the most fundamental" among all distributions.
...
- 850
4
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3
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428
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Maximum of independent, unit-variance Gaussians with non-zero means
Suppose $X_1,\dots,X_n$ are independent Gaussians, where $X_k \sim N(\mu_k,1)$. I am interested in
$$
Z \stackrel{\rm def}{=} \max_{1\leq k\leq n} X_k
$$
and specifically on the asymptotics of $\...
CommunityBot
- 1
4
votes
0
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190
views
Pedestrian proof of Gaussian chaos for order-two polynomial?
Let $\ell \geqslant 1$. Let us consider $(g_n)_{n \in \mathbb{N}}$ identically distributed independent real gaussian variables and real number $(a_{n_1,\dots n_{\ell}})_{(n_1, \dots, n_{\ell}s)\in\...
- 5,429
0
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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
1
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1
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512
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Conditions for Gaussianity of SDE
Fix $T>0$, $x \in \mathbb{R}^n$, and let $\mu$ and $\sigma_1,\dots,\sigma_m$ be (globally) Lipschitz-continuous functions from $[0,T]\times \mathbb{R}^n$ to $\mathbb{R}^n$. Thus, for every $0\leq ...
- 128k
2
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2
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690
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Concentration and anti-concentration of gap between largest and second largest value in Gaussian iid sample
Let $n \ge 3$ be an integer and let $X=(X_1,\ldots,X_n)$ be random vector with iid coordinates from $N(0,1)$. For $1 \le k \le n$, let $X_{(k)}$ be the value of the $k$th largest coordinate of $X$.
...
- 128k
0
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0
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64
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How to compute the following probability involving two normal random variables?
$\alpha$ and $\alpha'$ are two independent standard normal random variables. What's the conditional probability $$\mathbb{P}[\alpha >0, \alpha' >0|c_1<|\alpha - \alpha'|<c_2],$$ where $c_1$...
- 327
2
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3
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166
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On the probability of the multivariate normal with fixed pairwise correlations being coordinate-wise non-negative
This problem itself, admittedly, is not a research problem; but rather an intermediate step I've encountered in my research.
Let $(X_i:1\le i\le N)$ be a multivariate normal random vector where i) ...
- 128k
5
votes
2
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174
views
Integrability of Gaussian sums
Let $(X_1, \ldots, X_n)$ be a Gaussian vector, and $Z = \sum_{i=1}^n |X_i|$.
Since the map $x \mapsto e^{x^2}$, is convex, for any $t>0$
$$
e^{tZ^2} \, = \, e^{t \big(\sum_{i=1}^n |X_i| \big)^2}...
- 1
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1
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73
views
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 ...
- 4,943
0
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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 ...
- 6,853
0
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0
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320
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Does additive Gaussian noise preserves the Shannon entropy ordering?
Suppose that $Z$ is a Gaussian random variable independent of $X$ and $Y$. Moreover suppose that $h(X) \geq h(Y)$, where $h(\cdot)$ is the differential Shannon entropy.
Does relation $h(X+Z) \geq h(Y+...
- 63.9k
1
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1
answer
2k
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Convolution of two Gaussian mixture model
Suppose I have two independent random variables $X$, $Y$, each modeled by the Gaussian mixture model (GMM). That is,
$$
f(x)=\sum _{k=1}^K \pi _k \mathcal{N}\left(x|\mu _k,\sigma _k\right)
$$
$$
g(y)=\...
- 128k
1
vote
1
answer
116
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How to compute the following probability involving 4 normal random variables?
$\alpha, \alpha', \beta$ and $\beta'$ are four independent standard normal random variables, I am wondering how to compute the probability of the following two events:
$\alpha>\alpha'>0, \ \ \...
0
votes
0
answers
321
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Projecting a vector onto a random subspace
Let $A\in\mathbb{R}^{k\times d}$ be matrix with i.i.d. $\mathcal{N}(0,1/k)$ entries with $k<d$, and let $B=A^{\top}A$. I would like to compute the distribution of $Bx$ where $x\in\mathbb{R}^{d}$ is ...
- 129
5
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1
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224
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Hermite polynomial after rotation
When we consider the $n$-dimensional standard normal distribution, the orthogonal basis is $\{H_S(x)\}_{S}$ where $H_S(x) = \prod_{k=1}^n H_{s_k}(x_k)$. Here $H_*(x)$ is the normalized probabilist's ...
- 188k
0
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1
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194
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Gaussian integral $\int_X \|x\|_X^2 \mu(dx)$ in Banach space
For a centered Gaussian measure $\mu$ on a Hilbert space $X$, it is known that
$$\int_X \|x\|^2 \mu(dx) = tr(Q)$$ where $Q$ is the covariance operator. Is there a similar version for Gaussian measures ...
- 128k
3
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3
answers
501
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Identity on convolution with Gaussian measure
I've came across an identity once (I don't remember where) concerning convolutions of Gaussian measures. If I'm not mistaken, this identity was
\begin{eqnarray}
(\mu_{C}*f)(y) = \exp\bigg{[}\frac{1}{...
- 17.2k
0
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1
answer
209
views
Distribution of the direction of Gaussian random variable
Let $X$ be a complex normal random variable. (Or, equivalently, a 2D real normal.) Is it possible to say anything useful about the distribution of the phase of $X$? Is it possible to do estimation on ...
- 128k
12
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3
answers
3k
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Gaussian distribution, maximum entropy and the heat equation
I have asked this question on MathSE, but I got no replies, so I thought of trying here.
Consider the Gaussian distribution on $\mathbb{R}$ with mean $m$ and variance $t=\sigma^2$. This has the ...
- 5,380
6
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2
answers
904
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Gaussian measure on function spaces
I'm reading this classic work and I'd like to get deeper inside some of its techniques. In particular, the authors state: "We construct a Gaussian measure $d\mu_{0}(\phi)$ on a measure space of ...
- 34.7k
4
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2
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512
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Bounding an expectation involving i.i.d. standard Gaussians and Rademacher
I have tried to bound the following quantity, but cannot get the "right" (conjectured) bound:
$$
\phi(\gamma,d,n) = -1+e^{\frac{1}{2}n\gamma^2 d}
\mathbb{E}_{X}\left[\frac{\mathbb{E}_Z[\prod_{j=1}^n(...
- 1,372
5
votes
1
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1k
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A general formula for Gaussian integrals over matrix elements
The question I have is quite specific. So in the hope that this post might help others in the future, my problem boils down to solving the following integral:
$$I_\tau=\int \prod_{i, j=1}^{N} d J_{i ...
- 188k
3
votes
1
answer
2k
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Gaussian concentration inequality
Recently I found a concentration inequality for infinite dimensional Gaussian r.v.s in this paper. Specifically, Lemma 4 on page 307 states (without a proof) that
There exists a universal constant $...
- 128k
0
votes
1
answer
102
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Sign of expectation value
Consider a multivariate Gaussian-type measure $$d\lambda(x):=\nu_{\mu,\Sigma} e^{-\langle (x-\mu), \Sigma^{-1}(x-\mu) \rangle - \vert x \vert^2} $$
with vector $\mu \in \mathbb R^n$ and $\Sigma$ ...
- 17.2k
2
votes
1
answer
759
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History of the name "subexponential distribution" in probability
In probability theory, the term subexponential distribution has historically been used for a distribution whose CDF $F(x)$ satisfies the relation
$$
n(1-F(x)) \sim 1 - F^{*n}(x)
$$ for any $n \ge 1$ ...
- 128k
1
vote
1
answer
798
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Which distributions of $X$ and $Y$ yield a Gaussian $Z=XY$?
Let $Z=XY$ where $X$, $Y$ are random variables with support of non-trivial measure. For what distributions of $X$ and $Y$ can $Z$ be guaranteed to be Gaussian?
- 364
1
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1
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82
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Expectation value of multilinear forms over independent Gaussian vectors
Let $A$ be a symmetric multilinear form on $\left(\mathbb{R}^d\right)^{\otimes n}\times \left(\mathbb{R}^d\right)^{\otimes n}$ and consider the random variable:
\begin{align*}
X=A(g_1,\ldots,g_n,g_1,\...
- 128k
2
votes
0
answers
174
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Slepian's Lemma for Range?
Let $\vec{x}$ and $\vec{y}$ be zero mean $n$-variate Gaussian variables with covariances $\Sigma^x, \Sigma^y$. Suppose they have identical marginals ($\sigma_{i,i}^x = \sigma_{i,i}^y$ for all $i$), ...
- 620
1
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1
answer
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 ...
CommunityBot
- 1
4
votes
1
answer
431
views
Central limit theorem for resampling
This is a cross-post from stats.stackexchange.com. No answer has appeared there. Since this is a theoretical question, mathoverflow.net seems to be a more appropriate venue for it.
What is the analog ...
- 2,239
16
votes
6
answers
3k
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A normal distribution inequality
Let $n(x) := \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$, and $N(x) := \int_{-\infty}^x n(t)dt$. I have plotted the curves of the both sides of the following inequality. The graph shows that the ...
- 63.9k