All Questions
Tagged with pr.probability probability-distributions
1,386 questions
2
votes
1
answer
153
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A problem about normal distribution, independent random variables
Suppose $\alpha_1, ..., \alpha_n $ are independent identically distributed random variables, $ a_1, ..., a_n,b_1,...,b_n $ are non-zero constants. Is it true that if $ \sum_{i=1}^{n}a_i\alpha_i $ and $...
- 25.8k
2
votes
1
answer
114
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Does maximizing $D_u$ imply stochastic ordering?
Let $\mathscr P _0$ and $\mathscr P _1$ be two non-overlapping sets of probability distributions defined on $(\Omega,\mathcal{A})$. Consider the distance defined as $$D_u(P_0,P_1)=\int_\Omega \left(\...
- 8,071
1
vote
0
answers
265
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Time-inhomogeneous and state dependent Markov chain
We look at an inhomogeneous Markov chain $X_{n}$ that evolves according to the following transition probabilities:
$$
P(X_{n+1}=k+1|X_{n}=k)=\frac{f(k)}{n+1}\\
P(X_{n+1}=k|X_{n}=k)=\frac{n-f(k)}{n+1}\\...
- 11
5
votes
0
answers
204
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anti-concentration of multi-linear polynomials in Gaussian variables
A Gaussian variable $X_i\sim {\cal N}(0,1)$ is anti-concentrated in the following sense: for any $\epsilon>0$ we have:
$$
\mathbf{P}( |X_i| \leq \epsilon ) = O(\epsilon).
$$
Hence if we consider a ...
- 445
0
votes
1
answer
165
views
Efficiently Sampling of Multivariate Distributions in the Vicinity of a Manifold
I am given a multivariate distribution, that maps each point of $\mathbb{R}^n$ to its probability of being drawn as sample and, the convolution $\mathcal{S_r}\times\mathcal{M}$ of a manifold $\mathcal{...
CommunityBot
- 1
2
votes
1
answer
217
views
Measure space for trees and other algebraic datatypes
Given a measure space $\mathcal M$, I am wondering what kind of measure space $\mathcal T(\mathcal M)$ one could associate to the set of binary trees with elements from $\mathcal M$ at each node.
The ...
- 8,071
6
votes
0
answers
388
views
Closedness of a set of measures, where conditional marginals are in closed $\varepsilon$-ball w.r.t. Wasserstein distance
Let $(E,d)$ be a bounded polish space (separable, complete metric space satisfying $\sup_{x,y\in E} d(x,y) < \infty$). By $\mathcal{P}(E)$ we denote the space of Borel probability measures on $E$ ...
- 8,071
6
votes
0
answers
133
views
Random Balanced Assignment
A balanced assignment from from $N$ objects to $K$ classes is a mapping $\sigma\colon \{ 1, \ldots, N\} \rightarrow \{ 1, \ldots, K\}$ such that
$$
\textrm{Card}( \sigma^{-1} \{j \} ) = \textrm{Card} ...
2
votes
1
answer
369
views
Recovering a distribution from sample averages?
I'm working on a problem where I have $n^2$ real numbers $x_{11},...,x_{nn}$, all drawn i.i.d. from the same distribution $F$. I don't observe each $x_{ij}$, but I do observe the $n$ means:
$$\bar{x}...
CommunityBot
- 1
2
votes
1
answer
652
views
Concentration inequality for subgaussian^4
Let $X_1,...,X_N$ be IID, mean-zero random variables whose tail is bounded by a subgaussian-tailed variable to the fourth moment, i.e., for some $t \ge t_0 > 0$
$$
P(|X_i| > t) \le C\exp\left( -...
- 719
15
votes
2
answers
1k
views
Sum of independent random variables
We know that the sum of two independent normal random variables is again a normal random variable. But is the reverse right? If $X$ and $Y$ are independent random variables satisfying $X+Y$~$N(\mu,\...
- 91.8k
1
vote
4
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3k
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Differential Entropy of Random Signal
Prove that the Normal (Gaussian) Distribution with a given Variance $ {\sigma}^{2} $ maximizes the Differential Entropy among all distributions with defined and finite 1st Moment and Variance which ...
- 115
1
vote
1
answer
376
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Invariants on the space of probability distributions
There are many valuable functionals on the space of probability distributions which are invariant under variable transformations. (as an example KL divergence)
But all these functionals are defined ...
- 41
1
vote
1
answer
183
views
Diffuse measure space as a product of $[0;1]$ and another diffuse measure space
The title speaks of itself. How far is an arbitrary finite diffuse measure space from being almost isomorphic to a product of $[0;1]$ with another diffuse measure space? What would be reasonable ...
1
vote
1
answer
314
views
Solution of bimodal and multimodal Weibull distribution
Is there any closed form solution for $\sigma$ in a bimodal Weibull distribution function written in the following form:
$$ P(\sigma) = 1- exp\Bigg(-\alpha\Big(\frac{\sigma}{\sigma_1}\Big)^{m1} -\...
- 11.8k
-1
votes
1
answer
519
views
Poisson kernel is the Cauchy distribution, reference?
Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Can someone give me a reference to a proof that the Poisson kernel is the Cauchy distribution?
CommunityBot
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4
votes
2
answers
258
views
What theorem can be used to explain this occurrence?
I'm not highly versed in research-level mathematics. I do conduct research in cellular biology. I was wondering if you could help me find a term that can be referred to when discussing the following ...
- 34.7k
0
votes
0
answers
65
views
Wanted: example of a non-stationary sequence with reverse empirical measure
Assume we have a sequence $\xi=(\xi_1,\xi_2,\dots)$ of random variables such that $$\eta=\left(\frac{\sum_{i=1}^n \delta_{\xi_i}}{n}\right)_{n\geq 1}$$ is a reverse-martingale with respect to its own ...
- 211
7
votes
0
answers
3k
views
What is vague convergence and what does it accomplish?
For convenience, let's say that I have a locally compact Hausdorff space $X$ and am concerned with probability measures on its Borel $\sigma$-algebra $\mathcal{B}(X)$. Natural vector spaces to ...
CommunityBot
- 1
3
votes
0
answers
50
views
A transformation of a probabilistic distribution
Let $\mu$ be a probabilistic distribution on $\mathbb{N}$.
Consider the following random process $F_i(\mu)$.
First we choose $i$ numbers $x_1, \ldots, x_i$ randomly with respect to distribution $\...
- 2,576
4
votes
0
answers
103
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Does a non-exchangeable empirical reverse-martingale exist?
Consider a possible finite sequence $\xi_1,\xi_2,\dots$ of random variables and consider the measure-valued empirical process
$$\eta_n=\frac{\sum_{i=1}^n\delta_{\xi_i}}{n},\:\:\: n=1,2,\dots$$
Assume $...
CommunityBot
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3
votes
1
answer
345
views
Second moment of cos(x,y) for Normal x,y?
I'm trying to figure out second moment of the following quantity
$$y = \frac{\langle x_1, x_2 \rangle}{\left\|x_1\right\|\left\|x_2\right\|}$$
Where $x_1$, $x_2$ are sampled independently from $\...
CommunityBot
- 1
3
votes
0
answers
841
views
Is uniform distribution on unit sphere subgaussian?
Is uniform distribution on unit sphere subgaussian?
To be specific, let $X = (X_1,\dots,X_d) \sim \mbox{Unif}(\mathcal{S}^{d-1})$. What is the Orlicz-$\psi_2$ norm of $X$?
- 719
0
votes
1
answer
171
views
Distance of distributions of random variables, without PDF
Consider an interval $I$ with a smooth probability measure $d\mu (x) = c(x) dx$ and two known real measurable functions $f_1(x)$,$f_2(x)$. Both functions define a distribution on $X = {\rm Im} \, [f_1]...
- 3,574
4
votes
2
answers
475
views
midpoint between two normal distributions for the Rao-Fisher metric
Given two multivariate gaussian distributions $G_0 \sim N(\mu_0,\Omega_0)$ and $G_1 \sim N(\mu_1,\Omega_1)$, is there a closed-form formula for the gaussian distribution equidistant from them that is ...
- 3,574
5
votes
1
answer
1k
views
Supremum of a martingale
Let $(X_n)$ be a martingale. What can be said about the distribution of its maximum over a window of fixed length:
$$M_n = \max_{n-10 \leq k \leq n} X_k$$ or about the "range" over a window:
$$R_n = \...
CommunityBot
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2
votes
0
answers
107
views
Markov chain approximates a fractional diffusion
Let assume that
$$
dX_t=\mu(X_t)dt+\sigma(X_t)dW_t^H, X_0\in \mathbb{R}
$$
Where $\mu(.), \sigma(.)$ satisfy some conditions that guarantee $X_t$ exists, and $dW_t^H$ is a fractional Brownian motion ...
- 323
1
vote
0
answers
809
views
Generalized Chi squared distribution
What is the distribution of $Y Y^\top$ if $Y \sim N(\mu,\Sigma)$ is a generic multivariate gaussian vector?
- 719
7
votes
3
answers
346
views
Concentration Bound of $0/1$ permanent
If I pick a random $0/1$ $n\times n$ matrix with $0$ occuring with probability $p$ then what does the distribution of the permanent look like?
- 2,720
2
votes
0
answers
156
views
Sufficient condition for a solution to Hamburger moment problem
Let $\{m_n\}_{n=0}^{\infty}$ be a sequence of real numbers.
It is well known that there exist a positive Borel measure $\mu$ on the real line with moments given by $\{m_n\}_{n=0}^{\infty}$ if and ...
- 13.4k
0
votes
1
answer
201
views
The distribution of the maximum of a series of extreme value type I random variable
I have an infinite series of independent identically distributed random variables $\{X_i\}_{i=1}^\infty$ which follows extreme value type I distribution which can be found [here] (https://en.wikipedia....
- 24.8k
2
votes
0
answers
171
views
Distribution for the extreme values of a cumulative sum of normal variables
Suppose I have a sample $X$ of $n$ iid Normal random variables $(X_1,X_2,..,X_n)$.
Now, define the sample mean $u=\frac1n\sum_{i=1}^n X_i$ and let $Y_i=X_i-u$.
Let $Z_k=\sum_{i=1}^k {Y_i}$. Note ...
- 6,711
1
vote
1
answer
124
views
Reconstructing the number of distinct elements from a random projection
Assume we have an unknown sequence $x_1,\ldots, x_n\in \mathcal U$.
We get to observe the sequence $h(x_1),h(x_2),\ldots, h(x_n)$, where $h:\mathcal U\to \{1,\ldots, k\}$ is a random function such ...
- 618
2
votes
1
answer
352
views
Density of Non-Homogeneous Poisson Process
Given $\lbrace Y_i\rbrace$ a non-homogenous Poisson process with mean density $\theta y^{-1}e^{-y}$ where $y>0$ $(\theta>0)$. I.e., the number of points of $\lbrace Y_i\rbrace$ in $(a,b)$ with $...
- 188k
4
votes
2
answers
1k
views
Total progeny of a Galton-Watson branching process - standard textbook question
While analyzing some parallel-computing related algorithm, I came across a probability distribution with a particularly nice property (at least to me), but I am unable to write it down explicitly.
...
- 396
2
votes
1
answer
71
views
Distances between probability distributions by the variance of the test functions
Let $P$ and $Q$ be two probability distributions on $\mathbb{R}$. The goal is to obtain a notion of ``distance'' between $P$ and $Q$, e.g., total variation distance, K-L divergence.
Let $f\colon \...
- 29.8k
0
votes
0
answers
114
views
Merging Poisson/lognormal processes
We know that merging two Poisson processes results in another Poisson process with a rate that is the sum of the two original rates.
(https://www.probabilitycourse.com/chapter11/...
3
votes
3
answers
292
views
A question in central limit theorem
Suppose $\{X_n,n\ge1\}$ are independent r.v., $E(X_n)=0$, $\operatorname{Var} \left(X_n\right)=\sigma_n^2<\infty$. Set $S_n=\sum_{i=1}^nX_i$ and $s_n^2=\sum_{i=1}^n\sigma_i^2$, assume
$$\frac{S_n}{...
- 211
3
votes
0
answers
82
views
For a given Gaussian vector, which rectangular parallelepiped with unit volum has the largest probability?
Let $X$ be a centered Gaussian vector of $\mathbb{R}^n$ and $\Gamma$ its covariance matrix. We assume that diagonal coefficients of $\Gamma$ are all equal to 1.
We are looking for a rectangular ...
- 3,085
1
vote
0
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49
views
A question about the prediction error
I am reading about the prediction error estimation and I found the following:
Suppose we have ${\mathbf{Y}}=\mathbf{x}_0+ \epsilon$, where, $\epsilon$ is normally distributed as $\sim \mathcal{N}(0, \...
- 11
0
votes
0
answers
57
views
Parametric distribution where the parameter follows a diffusion process
I'm looking for a distribution $P_{\theta}$ with pdf $f (t,\theta)$ over $\mathbb{R}^{+}$ such that there exists functions $\mu(\theta)$ and $\sigma(\theta)$ such that for all $t>0$:
$$\mu(\theta)\...
- 1,902
3
votes
1
answer
190
views
Solution for Moment problem
I want to invert a sequence of moments and find a function f(x) satisfying:
$$m_r=\int x^{r}f(x) dx=\int x^{r} dF(x)$$
The sequence of moments is given by:
$m_{2s+1}=0$
$m_{2s}=\sum_{k=1}^{s}\binom{...
- 60.6k
0
votes
1
answer
151
views
Can an unskewed distribution be expressed as product of a normal and another distribution?
Let $x$ be a continuous random variable with zero mean and zero skew. What are the conditions under which we can say that $x$ can be expressed as the product $z y$ where $z$ is a standard normal and $...
- 54.2k
6
votes
1
answer
777
views
Stein's Lemma for Discrete Distribution
Stein's Lemma in its standard form states that $X \sim N(0,1) \Leftrightarrow E[f'(X) - X f(X)] =0 $ for all bounded one-time differentiable functions $f$ (I am ignoring the exact conditions on $f$ ...
- 8,071
3
votes
0
answers
267
views
Conditional distributions of uniformly distributed random orthonormal matrices
Let $U, U'\in R^{d\times k} (d>k)$ be two independent uniformly distributed random orthonormal matrices. In specific, let $S$ be the set of all $d\times k$ orthonormal matrices. Here 'uniform' is ...
- 1,127
10
votes
1
answer
263
views
q-versions of the geometric distribution and their names
I'm trying to set straight various $q$-deformations of the standard geometric distribution.
The geometric distribution on $\left\{ 0,1,\ldots \right\}$ is well-known, it has
$$
\mu_1(X=j)=(1-p)p^j,\...
- 1,765
1
vote
0
answers
575
views
Bounding the total variation distance of two specific random variables?
Let $X$ and $Y$ be two independent discrete random variables, and $Z$ be a function of $X$ and $Y$, i.e., $Z=f(X,Y)$. Suppose that $\Gamma$ is a set such that
$$\mathrm{Pr}[(X,Y)\in\Gamma]\geq 1-\...
- 287
1
vote
0
answers
43
views
Distribution of maximum minor of a random matrix with one special column
Given $m,n,\ell\in\Bbb N$ and $\beta\in(0,1)$ consider the uniformly picked random matrix $A\in\Bbb Z^{n\times (n+1)}$ with $0\neq|\mathsf{det}(A^\circ)|\leq m^{\frac 1\ell}$ where $A^\circ$ is the ...
- 13.9k
8
votes
2
answers
633
views
Maximal entropy distribution with given conditionals
It is well known that of all the joint distributions $p(x,y)$ with fixed marginals $p(x),p(y)$, the one with the highest entropy is:
$$
p(x,y)=p(x)p(y).
$$
Suppose instead that we have conditionals. ...
CommunityBot
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3
votes
0
answers
75
views
Covariance of censored/clipped Gaussians
I am interested in the covariance of two clipped (or censored) Gaussian variables.
More precisely, let $g_1 \sim N(0,\sigma_1^2)$ and $g_2 \sim N(0,\sigma_2^2)$ be two (dependent) Gaussians with $\...
CommunityBot
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