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.

**0**

votes

**0**answers

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))]...

**0**

votes

**0**answers

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**

votes

**0**answers

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 ...

**0**

votes

**2**answers

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**

vote

**0**answers

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(\...

**0**

votes

**0**answers

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

**0**answers

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**

vote

**0**answers

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**

votes

**0**answers

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

**1**answer

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?

**0**

votes

**1**answer

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, ...

**0**

votes

**0**answers

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 ...

**0**

votes

**0**answers

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

**2**answers

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'}...

**-1**

votes

**0**answers

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 $n > 0$, $p, q \in [0, 1]$, ...

**1**

vote

**1**answer

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 ...

**0**

votes

**0**answers

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**

vote

**0**answers

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 ...

**1**

vote

**1**answer

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\...

**1**

vote

**0**answers

47 views

### Families of distributions with a certain symmetry property?

Consider the probability distribution $\mathcal{N}_n$ on $\mathbb{R}^n$ whose density is $$(2\pi)^{-n/2}e^{-\frac{1}{2}||\vec{x}||^2} = \prod_{i=1}^n \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}x_i^2}$$
...

**6**

votes

**4**answers

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

**3**answers

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-...

**1**

vote

**2**answers

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

**1**answer

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**

vote

**1**answer

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

**1**answer

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 ...

**0**

votes

**1**answer

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

**1**answer

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 $l$-...

**0**

votes

**1**answer

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

**3**answers

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 ...

**1**

vote

**0**answers

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 ...

**1**

vote

**0**answers

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_{...

**0**

votes

**2**answers

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_{...

**0**

votes

**0**answers

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

**3**answers

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

**2**answers

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

**1**answer

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

**1**answer

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 ...

**1**

vote

**1**answer

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**

votes

**1**answer

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 ...

**1**

vote

**0**answers

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

**1**answer

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

**1**answer

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 \...

**1**

vote

**0**answers

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

**1**answer

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**

vote

**0**answers

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 $X$...

**0**

votes

**0**answers

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**

vote

**0**answers

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**

vote

**1**answer

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 ...

**0**

votes

**0**answers

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....