In probability and statistics, a probability distribution assigns a probability to each measurable subset of the possible outcomes of a random experiment, survey, or procedure of statistical inference.
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36 views
Upper bound on expectation of product
I want to upper-bound the following quantity:
$$\mathbb{E}_Y\left[f(Y)g(Y)\right] $$
The idea would be to get something of the shape: $\mathbb{E}_Y[f(Y)]\cdot h(Y)$
where $h(Y)= j(\mathbb{E}_Y[k(g(Y))]...
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0answers
98 views
Which probability distribution has the most outliers?
Let $k$ be a positive real number. Which probability distribution over $\mathbb R$ maximizes $P(|x-E(x)|>k\cdot \operatorname{std}(x))$?
-1
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0answers
36 views
Understanding a probability formula of a random walk [on hold]
I have the following problem: Let's assume $G$ is a graph with vertices in red or blue colour. There is no limitation on how we connect the vertices, i.e., a red vertex can be connected with either ...
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2answers
90 views
Distribution of the individual coordinates of a uniform random vector on a high-dimensional sphere
Let $X=(X_1,\ldots,X_n)$ be a random vector uniformly distributed on the $n$-dimensional sphere of radius $R > 0$. Intuitively, i think that for large $p$ every coordinate $X_i$ is normally ...
1
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0answers
25 views
Choice of residual function for least squares error minimization
Good morning,
I have the a set of data $(\sigma,D,\alpha_0)_i$, $i=1...n$ data.
I want to determine two parameters $K_{IC}$, $C_f$ in the basic equation given as
$K_{IC} = \sigma \sqrt{D} k_0(\...
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0answers
24 views
How can I solve a constrained optimization problem with a random number of decision variables?
Here is my problem. Let $A_t$ be a random variable with Poisson-Binomial distribution with set of success-probabilities $\{q_1^{(t)},\ldots,\,q_n^{(t)}\}$, with $t\in\{1,\,2,\,3,\ldots\}$, $n\in\...
2
votes
0answers
92 views
Random walk and comparing sums of Exponential random variables
Let $\sigma$ be the time a nearest neighbor random walk started at 1 that has probability $p>1/2$ of moving left reaches $0$. Let $\sigma'$ be an independent copy of $\sigma$. Let $(X_k)_1^\infty$ ...
1
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0answers
46 views
What are the moments of Kolmogorov Complexity for a Random Variable?
Given a random variable $X$ distributed under some computable distribution $P$ we have,
$$0 \le E[K(X)] - H(P) \le K(P)$$
Where $H(P)$ is the entropy of $P$. I tried using Hoeffding concentration ...
0
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0answers
62 views
Asymptotic distribution of $n\mathbb E_{\hat{P}_n}[g(Z;\theta)]^T\operatorname{Cov}_{\hat{P}_n}[g(Z;\theta)]^{-1}\mathbb E_{\hat{P}_n}[g(Z;\theta)]$
Setup
This question is a followup on this question. I'm interested in the asymptotic distribution of certain quadratic forms.
So, let $Z$ be a $p$-dimensional random vector with (unknown) ...
1
vote
1answer
64 views
Generalization of inverse transform sampling
If X is a random variable over an arbitrary alphabet, is there a (deterministic) function f() such that X = f(U), where U is a uniform random variable over the unit-interval?
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1answer
24 views
Cumulative Order Statistics of Independent Non Identical Distributions
I understand that the p.d.f of order statistics for Independent Non Identical Distributions are given by the Bapat-Beg theorem as previously explained in another question. As explained in the article, ...
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0answers
25 views
Rate of convergence of centered Hotelling's statistic to Chi-squared distribution
Consider the Hotelling's statistic $H_n := n\mu_n\Sigma_n^{-1}\mu_n$, where $\mu_n$ (resp. $\Sigma_n$) is the empirical mean (resp. empirical covariance matrix) of a zero-mean random $d$-dimensional ...
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0answers
27 views
Distribution of an arithmetic progression with respect to a monotonic sequence with distributed increment
Let $T>0$, $\Delta>0$, $\Delta<T$,
$\tau : N_e \rightarrow (0,\infty)$ be a sequence uniformly distributed on
the interval $[T-\Delta,T+\Delta]$. Here $ N_e $ is the set of non-negative ...
3
votes
2answers
209 views
Effect of perturbing the atoms of a measure on the Wasserstein distance
Let $(X,d)$ be a metric space, $x_1,\ldots,x_N\in X$ and $x_1',\ldots,x_N'\in X$ be atoms, and $G=\sum_{i=1}^Np_i\delta_{x_i}$, $G'=\sum_{i=1}^Np_i'\delta_{x_i}$, and $G''=\sum_{i=1}^Np_i'\delta_{x_i'}...
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0answers
20 views
A simple function that preserves the ordering of Binomial CDF
I am looking for a simple function $f(n, p, q)$ that exactly preserves the ordering of the cumulative probabilities $\mathbb{P}(\rm{Binom}(n, p) > n q )$, for all $n > 0$, $p, q \in [0, 1]$, ...
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1answer
136 views
continuity entropy with respect gibbs measures
Let $X=\{0,1\}^{\mathbb{N}}$. For simplicity I consider Bernoulli measures on $X$ only.
Let $f:X\to \mathbb{R}$ be Holder continuous. The measure $\mu$ is a Gibbs measure with potential $f$ if there ...
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0answers
32 views
Expression for the Markov Chain CLT variance for an arbitrary initial distribution
Let
$(\Omega,\mathcal A,\operatorname P)$ and $(E,\mathcal E,\pi)$ be probability spaces
$(X_n)_{n\in\mathbb N}$ be a $(E,\mathcal E)$-valued stochastic process on $(\Omega,\mathcal A,\operatorname P)...
1
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0answers
25 views
Expected value of inverse of complex non-central Wishart matrix
I have a matrix $W$ that abides a complex non-central Wishart distribution.
My question is what the expectation of the inverse is, i.e., how to compute
$$\mathbb{E}(W^{-1}).$$
I have tried to read up ...
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1answer
132 views
Asymptotic distribution of $\mathbb E_{\hat{P}_n}[Z]^T\operatorname{Cov}_{\hat{P}_n}[Z]^{-1}\mathbb E_{\hat{P}_n}[Z]$
Under very general conditions on the random $p$-dimensional vector $Z$, what can be said about the asymptotic distribution of the (random) scalar quantity $R_n := \mathbb E_{\hat{P}_n}[Z]^T\...
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0answers
47 views
Families of distributions with a certain symmetry property?
Consider the probability distribution $\mathcal{N}_n$ on $\mathbb{R}^n$ whose density is $$(2\pi)^{-n/2}e^{-\frac{1}{2}||\vec{x}||^2} = \prod_{i=1}^n \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}x_i^2}$$
...
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4answers
257 views
Improvement of Chernoff bound in Binomial case
We know from Chernoff bound
$P\bigg(X \leq (\frac{1}{2}-\epsilon)N\bigg)\leq e^{-2\epsilon^2 N}$ where
$X$ follows Binomial($N, \frac{1}{2}$).
If I take $N=1000, \epsilon=0.01$, the upper bound is ...
3
votes
3answers
172 views
What is the probability distribution of the $k$th largest coordinate chosen over a simplex?
Suppose we're selecting points uniformly at random from the $N$-simplex
$S_N = \{x \in \mathbb R^{N+1}: $ all $ x_i \ge 0$ and $x_1 + \ldots x_N = 1\}$.
One way to do this in practice is choose $N-...
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2answers
108 views
lower bound the probability of at least L collisions
Lets say we get a list $M$ containing $|M|=\sqrt{L\cdot N}$ randomly and independtly drawn elements from a set of size $N$. And lets denote the $i$-th element of the list $M$ by $M[i]$.
If we now ask ...
2
votes
1answer
69 views
Controlling Mean Difference Between Product and Joint Distributions Using Optimal Transportation
Suppose we have nonindependent random variables $X \sim P$ and $Y \sim Q$, where $P$ and $Q$ denote their marginal distributions. We are interested in upper bounding
$$
|\mathbf{E}_{X, Y\sim P \...
1
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1answer
33 views
Asymptotic eigenvalue distribution of sum of two i.i.d random matrices with Marchenko Pastur distributed eigenvalues?
Is there a method using random matrix theory and NOT using free probability to determine the asymptotic eigenvalue distribution of the random matrix $\mathbf{M}=\mathbf{X}_1+ \mathbf{X}_2$? where:
$\...
3
votes
1answer
108 views
Log concavity of the maximum of dependent Gaussians
Let $Z_1,\dots,Z_n$ be dependent Gaussian random variables. Is it true that $X=\max\{Z_1,\dots,Z_n\}$ has a log-concave distribution function? This is true for the independent case, but is it true in ...
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1answer
87 views
Is the normal product distribution sub-gaussian?
Consider the normal product distribution, which is the distribution of the product of two or more independent normal variables. Particulary, focus in the case where the multiplied normal variables are ...
3
votes
1answer
114 views
Tail probability of random projection
Suppose $v\in R^n$ is a constant unit vector. $P_l$ is a random projection matrix to an $l$ dimensional subspace of $R^n$ which is uniformly sampled from $G(l,R^n)$ which is the collection of all $l$-...
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1answer
56 views
Bivariate Poisson-Binomial distribution
Suppose you have $100$ coins whose probabilities of obtaining the outcome "head" are $p_1,\ldots,\,p_{100}$. These probabilities are not necessarily equal each other. Consider the following random ...
7
votes
3answers
261 views
Distribution of sum of two permutation matrices
Determinant and permanent of sum of two $n\times n$ permutation matrices can be arbitrarily different.
What is the distribution of determinant of sum and difference of two $n\times n$ permutation ...
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0answers
42 views
Compare KS test and Wasserstein distance or Earth mover's distance
I have tried the following question in couple of exchange sites but I did not get any views or reply. I am asking here as I am kind of desperate for the answer, please be considerate. Any suggestion ...
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0answers
49 views
Closed-form formula for Wasserstein distance between uniform discrete distribution and discrete distribution with same support
Let $x_1,\ldots,x_n$ be $n \ge 1$ distinct points in $\mathbb R^d$ and consider two discrete distributions on these points $\mu = (1/n)\sum_{i=1}^n\delta_{x_i}$, and $\nu = \sum_{i=1}^n\nu_i\delta_{...
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2answers
54 views
Use Importance sampling for multimodal and multivariate distribution draws, how to choose proposal distribution?
I'm in trouble trying to generate samples following a particular distribution which is not numerically known perfectly. Let us consider a $R^n$ space provided with an orthonormal base $( e_{1},...,e_{...
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0answers
25 views
Conditional distribution of maximum of the multivariate normal distribution
Suppose $X=[X_1,...,X_n]^T$ follows an $n$-dimensional multivariate normal distribution $\mathcal{N}_n(\mu_1,\Sigma_1)$ and $Y=[Y_1,...,Y_n]^T$ follows $\mathcal{N}_n(\mu_2,\Sigma_2)$, and $Y$ is ...
10
votes
3answers
503 views
Entropy and total variation distance
Let $X$, $Y$ be discrete random variables taking values within the same set of $N$ elements. Let the total variation distance $|P-Q|$ (which is half the $L_1$ distance between the distributions of $P$ ...
4
votes
2answers
241 views
The probability density function of the number of coins to first fill one box of $N$
Given $N$ boxes with the same capacity $C$, I toss coins into the boxes uniformly, one by one. When any one of the boxes is full, the sum of the coins in all boxes is denoted $S$. How to compute the ...
2
votes
1answer
189 views
A modest generalization of the law of large numbers
Suppose I collect $2n$ independent samples of a probability density function $f$, which are separated into pairs $\{X_i^1, X_i^2\}$ for $1\leq i\leq n$. Suppose I now consider the set of all $2^n$ ...
3
votes
1answer
52 views
Lower bound on the sum of pmf squared of a hypergeometric distribution
I am working on a proof of correctness for an algorithm I came up with. I encountered the following problem en route. I would appreciate if anyone had some idea or could point me to the relevant ...
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1answer
80 views
What is the order of the left tail of a mixture of non-central chi-square?
Let $\mu\sim N(0,1)$, $Z\sim N(\mu,1)$. Then $Z$ can be viewed as a mixture of Gaussians. It can also be viewed as a Gaussian but there is a prior for the mean.
Let $X\sim\exp(\lambda)$ where the ...
3
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1answer
202 views
Updating Geman and Geman (1984) on image restoration
I am reading the seminal paper
Stuart Geman and Donald Geman, Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images, IEEE Transactions on Pattern Analysis and Machine ...
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0answers
25 views
Supremum of expectations over a family distributions close to a base distribution
Let $p$ be a probability distribution on a space $X$ , $f:X \rightarrow \mathbb R_+$ be a measureable function, and $0 < \alpha \le 1 \le \beta < \infty$. Define a sub-collection $\mathcal Q \...
4
votes
1answer
131 views
General distributions with the “transportation-cost inequality” property to piece log-concave distributions
It is now known [Otto et Villani 2000; Cordero et al 2006; etc.] that on an $n$-dimensional smooth Riemannian manifold $X$ and a probability measure $\mu$ on $X$ with density $d\mu \propto e^{-V}dvol$ ...
3
votes
1answer
186 views
wasserstein distance between distributions with bounded ratio
Let $p$ and $q$ be probability distributions on a metric space $X=(X, d)$ with densities $dp$ and $dq$, such that there exists $0 < \alpha < \beta < \infty$ satisfying
$$
\alpha d p \le dq \...
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0answers
27 views
Singular values of random matrices with inhomogeneous variances
If $X$ is a random rectangular matrix with independent identically distributed entries of zero mean and equal variance, then as $X$ gets big its singular values tend to a Marchenko-Pastur distribution....
2
votes
1answer
82 views
Stochastic domination of Gaussian random vectors
Let $S$ be the class of all $2$ by $2$ matrices of the form
$$\begin{bmatrix}
1 & a \\
a & 1
\end{bmatrix},\, |a|\leq 1.$$
Is there a single matrix $M\in S$ such that for ...
1
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0answers
56 views
Upper bound for $\sup_{W_d(Q, P) \le \epsilon} Q(A)$, where $W_d$ is the Wasserstein metric
Let $X=(X,d)$ be a metric space and let $W_d$ denote the Wasserstein metric induced by this metric, on the space of probability distributions on $X$. Let $\epsilon \ge 0$, $A$ be a Borel subset of $X$...
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0answers
31 views
General conditions for Gaussian isoperimetric inequaliteis
Context
I'm doing some work in which i need to show that the blow-up $B_\epsilon$ of a Borel set $B$ has large measure for $\epsilon$ sufficiently large. I've found a very attractive tool for doing ...
1
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0answers
29 views
Probability of exiting on the boundary for a monotone Lévy-type process
Let the continuous function $\ell:\mathbb R \times(0,\infty)\to[0,\infty)$ be a Lévy-type kernel, such that
$$
\sup_{x}\int_0^\infty \min\{1,y\}\ell( x, y)\,dy<\infty,
$$
and suppose that $\...
1
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1answer
79 views
A distribution which is Wasserstein-close to a compactly supported distribution is almost compactly supported
I wonder whether this is true: If a distribution is very close (in the Wassertein sense) to another distribution with compact support, then the former must put only tiny amount of mass outside a ...
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0answers
22 views
Approximate in $W_1$ sense, an empirical distribution with restriction of true distribution on a set
Let $\mu$ be a probability distribution on a metric space $X=(X,d)$ (to avoid unnecessary complications, assume the full filtration $2^X$) and let $x_1,x_2,\ldots,x_N$ be a sample of size $N$ drawn i....
