All Questions
Tagged with pr.probability probability-distributions
1,384 questions
3
votes
0
answers
98
views
Probability measure on $\mathbb{R}^n$ with given marginals and given correlation matrix
In all what follows, let $\mathcal{P}(\mathbb{R}^n)$ denote the set of probability measures on $(\mathbb{R}^n, \mathcal{B}(\mathbb{R}^n))$ and $\mathcal{C}_n$ the set of $n \times n$ correlation ...
- 279
1
vote
1
answer
92
views
Geometric sampling problem in the Euclidean space in high dimensions
Let $T$ be the triangle whose vertices are three given points $\mathbf{x}, \mathbf{y}, \mathbf{z}\in\mathbb{R}^d$.
Question: What computationally efficient strategy can we use to sample a point $\...
- 1,677
0
votes
1
answer
266
views
CDF of a log-concave discrete random variable
In the continuous setting, it's known that if a density function is log-concave , then its CDF is also log-concave.
My questions:
What can we say about this in the discrete setting?. For ex: Is the ...
- 128k
2
votes
0
answers
83
views
Concentration inequalities for sets
Assume that we have a random set $B$ which is constructed by selecting elements from $U = \{ X_1, \dots, X_n \}$ where $X_i$ are independent samples from Gaussians with means $\mu_i$ and variances $\...
- 143
21
votes
1
answer
32k
views
How to compute KL-divergence when PMF contains 0s?
From the Wikipedia page on Kullback-Leibler divergence, the way to compute this metric is to utilize the following formula:
The way I understand this is to compute the PMFs of two given sample sets ...
10
votes
2
answers
488
views
A functional equation involving the inverse function
$\newcommand\ep\epsilon\newcommand\R{\mathbb R}$Let $P$ denote the set of all continuous probability density functions (pdf's) $p$ on $\R$ vanishing at $\pm\infty$. Let us say that a pdf $p\in P$ is ...
- 61.9k
0
votes
2
answers
80
views
Name of distribution of the parameter of a Poissonian
Consider a Poisson process $\hat{n}$ with with parameter $t$ and distribution
$$f_t(n) = e^{-t} \frac{t^n}{n!}$$
Now instead suppose to have a random variable $\hat{t} \in \mathbb{R}^+$ whose ...
- 3,065
0
votes
1
answer
188
views
Asymptotic behavior of the Student's t-quantile function of Student's t-cumulative distribution function
Let's denote
$F_{t_u}^{-1}(x)$ the quantile function of the Student's t-distribution $t_u$ with $u$ degrees of freedom and
$F_{t_v}(x)$ the cumulative distribution function of the t-distribution $t_v$...
- 128k
2
votes
1
answer
268
views
Union bound probability of random union
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. Let $\{E_i\}_{i = 1}^N,$ with $E_i \in\mathcal{F}$ be a set of events and let $i(X)$ be a R.V. assuming values in $\{1,...,N\}$
Is there ...
- 128k
2
votes
2
answers
511
views
Geometry interpretation of any continuous random variable
Given a continuous random variable $X$ with the cdf $F_X(x)$, I want to know whether there exists a random vector $\mathbf{Z}$ uniformly distributed in a geometry region $\mathscr{Z}_n$ in $\mathbb{R}^...
- 128k
2
votes
1
answer
241
views
Weak continuity of law
Let $\mathcal{P}_2(\mathbb{R}^n)$ denote the set of all Borel probability measures on $\mathbb{R}^n$ with finite variance and weak topology. Let $X_t$ be a strong solution to the SDE with initial ...
- 5,405
1
vote
0
answers
75
views
Why does Y. Moshe Vardi use this specific matrix when estimating source-destination traffic intensities with EM algorithm?
Sorry for the verbose title, but the question is super specific. If you happen to know a site better suited for these types of question, feel free to direct me.
The article to which I am referring to ...
- 35.5k
1
vote
2
answers
197
views
Sampling method for a specific distribution in high dimensions
We are given a set $X$ of $n$ points $\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n\in\mathbb{R}^d$, where $d\ll n$. Given any point $\mathbf{p}$ on the unit $(d-1)$-sphere $\mathcal{S}$, we define
...
- 1,677
0
votes
0
answers
144
views
Optimization over the set of all bounded probability measures
Given $X$ finite, fix a continuous function $\theta \in \Delta^+ (X) \to [0,1]$, fix a probability measure $\mu^*$, and a $\varepsilon > 0$. Consider:
$$
\max_{\mu \in \Delta^+ (X)} \theta (\mu), \...
- 67
0
votes
1
answer
651
views
Stable law and the domains of attraction
The multivariate generalised central limit theorem with their domains of attraction was given by Rvačeva (see also this post). The original paper is not very accessible on the internet, and neither ...
- 380
1
vote
1
answer
157
views
Constructing representations of probability revision functions
Let $P$ be a probability distribution over a finite Boolean algebra $\mathfrak{B}$, and fix a parameter $t_{P} \in (\frac{2}{3}, 1)$. Define the `revision function of $P$', $R_{P}: \mathfrak{B}\...
- 631
0
votes
1
answer
141
views
Arbitrarily bad rates of convergence in Wasserstein metric
Suppose $W_p(\mu_n,\mu)\to 0$ and $d(E(\mu_n),E(\mu))<r_n$. Here, $W_p$ is the $p$th-order Wasserstein distance (with respect to the metric $d$) and $\mu_n,\mu$ are probability measures on some ...
- 128k
0
votes
1
answer
268
views
Tightness on a set $A$ implies tightness on a set $B$ where $A\subset B$?
From the book Billingsley - Convergence of probability measures, 1999, we have the following definitions of tightness and relative compactness and the Prohorov's theorem:
Tightness: Let $\Pi$ be a ...
- 128k
0
votes
2
answers
874
views
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
vote
3
answers
536
views
On exponential distributions and dot products
Let
$a, b$ be two variables drawn from an exponential distribution with parameter $\lambda_1$.
$c, d$ be two variables drawn from an exponential distribution with parameter $\lambda_2$.
I am ...
- 3,486
1
vote
0
answers
68
views
(Anti-)concentration of gap between largest and second largest component of multivariate random gaussian vector
Let $n$ be a large positive integer and let $Y=(Y_1,\ldots,Y_n)$ be a zero-centered random $n$-dmensional real vector with covariance matrix $\Sigma$, an $n$-by-$n$ positive definite matrix with ...
- 6,853
0
votes
1
answer
967
views
Bound the norm of sum of random vector that generated from standard basis
I have a question like this:
Consider $N$ samples $X_1, X_2, ..., X_N$ that uniformly random generated from standard basis $\{e_i, i=1,2,...,d\}$, i.e. $(1,0,0,\cdots,0),(0,1,0,\cdots,0),(0,0,1,0,\...
- 128k
2
votes
1
answer
102
views
If signed measures $\mu_n$ are such that $\mu_n\to\mu$ and $\|\mu_n\|\to c\in(0,\infty)$, does $\exp^*(\mu_n)/\|\exp^*(\mu_n)\|$ necessarily converge?
$\newcommand{\R}{\mathbb R}$Let $M$ denote the set of all finite signed measures on a separable Banach space $B$. For any $\mu\in M$, let
\begin{equation*}
\exp^*(\mu):=\sum_{k=0}^\infty\frac{\mu^{...
- 128k
10
votes
4
answers
681
views
The min of the mean of iid exponential variables
Let $X_1, \ldots, X_n, \ldots$ be iid exponential random variables with mean 1. It is well-known that $\min_{1\le j < \infty} \frac{X_1 + \cdots + X_j}{j}$ follows the uniform distribution U(0,1). ...
- 3,255
-2
votes
1
answer
108
views
If a sequence of measures is weakly convergent outside each compact ball, the sequence itself is weakly convergent
Let $E$ be a $\mathbb R$-Banach space and $\mathcal M_+(E)$ denote the space of finite nonnegative measures on $\mathcal B(E)$.
If $\lambda\in\mathcal M_+(E)$, let $$\left.\lambda\right|_\delta(B):=\...
- 29.8k
1
vote
1
answer
283
views
Bounding the probability Jaccard distance with total variation distance
Let $\Delta_n$ be the set of all probability vectors on $n$ points, also known as the $n$-simplex. Let $x, y \in \Delta_n$ be two probability vectors, that is, $\sum_{i=1}^n x_i = 1$ and $x_i \geq 0$, ...
- 4,466
0
votes
1
answer
294
views
Joint distribution of random Fourier coefficients
Consider choosing a Boolean function $f : \{0, 1\}^{n} \rightarrow \{-1, 1\}$ uniformly at random from the set of all Boolean functions and consider the random variable $\left(\hat f(z_{1}), \hat f(z_{...
- 128k
0
votes
1
answer
582
views
Integrability of $\int \log(f(x)) f(x) dx$ for a probability density function $f$
I am looking for weak conditions when a probability density function $f$ on $\mathbb{R}^d$ has a finite integral
$$
\int_{\mathbb{R}^d} \log(f(x)) f(x) dx.
$$
Any references would be appreciated.
- 128k
1
vote
1
answer
135
views
Probabilistic problem on the covariance matrix of a multivariate normal distribution [closed]
We have a random variable $\mathbf{X}\sim\mathcal{N}_d(\mathbf{\mu},\mathbf{\Sigma})$, where $\mathcal{N}_d(\mathbf{\mu},\mathbf{\Sigma})$ is a $d$-dimensional multivariate normal distribution with ...
- 128k
7
votes
2
answers
873
views
Which random variables can be written as the difference of two independent positive random variables?
Can we characterize random variables $X$ that satisfy
$$
X\sim Y - Z
$$
for two independent positive random variables $Y$ and $Z$?
Are $Y$ and $Z$ unique in some sense?
Can (one possible choice of) $Y$...
- 24.2k
1
vote
1
answer
517
views
log-like distance between probability distributions
Given two probability density functions (PDF) $f$ and $g$, both defined over the same set $X$, there are many ways to describe/measure the distance between them, e.g., KL divergence and Hellinger ...
- 3,979
1
vote
2
answers
163
views
Coupling a binomial - parity conditioning
If I have a binomial $X \sim B(n,p)$, and another binomial $X' \sim B(n,p)$ conditioned on $X'$ being of even parity. Is it true that there always exists a coupling for $(X,X')$ with $|X-X'| \le 1$? (...
- 128k
5
votes
0
answers
130
views
Random process on a sequence of rolls of an $n$-sided die
Let $\ X:=X_{k\,n}\ $ be a random variable of a $n$-sided die where $\Pr(X=i)=\frac{1}{n}$ for each $i\in\{1,2,\ldots,n\},\ $ where $\ k\in\{1, 2, \ldots,n\}\ $ and $\ n\ $ are fixed. Let $t$ be a ...
- 83
2
votes
0
answers
164
views
Finding an optimal strategy for a combinatorial sequential game
We are given a set $\{p_1, p_2, \ldots, p_n\}$ of players and a set of $\{\ell_1, \ell_2, \ldots, \ell_n\}$ of locations, where $n\in\mathbb{N}$. Each location can be either free or occupied, and each ...
- 83
1
vote
3
answers
339
views
Coupon collector targeting a collection among many
I am interested in the following problem:
We are given a universe $U$ of $n$ coupons, partitioned into $k$ collections, $C_1,\dots C_k$.
At each time step $t$, a coupon $X_t$ is selected uniformly at ...
- 4,361
0
votes
1
answer
70
views
Simulation of multivariate logistic distribution conditional to a plane
For an algorithm, I have to simulate $X_1, \ldots, X_n \sim_{\text{iid}} \text{Logistic}(0,1)$ conditionally to the event $(X_1, \ldots, X_n) \in P$, where $P$ is an affine plane in $\mathbb{R}^n$.
I ...
- 128k
0
votes
1
answer
129
views
Obtaining Chebyshev bound based thresholds for a particular tail probability using higher order moments
This is a research question. Consider an univariate non-negative random variable $q$. I intend to have a desired tail probability, say $\mathcal{A}$. If I don't know the distribution of $q$, but I ...
- 3,937
1
vote
1
answer
118
views
Laplace transform of the product of two gammas
Suppose that $X$ and $Y$ are both gamma distributed with shapes $a,b$ and unit scales / unit rates. To fix ideas, X has Laplace transform given by:
$$L_X(t) = \mathbb{E}(e^{-tX}) = (1 + t)^{-a}$$
How ...
- 188k
2
votes
1
answer
218
views
Probability distribution optimization problem of distances between points in $[0,1]$
Let $\mathcal{D}$ be a probability distribution with support $[0,1]$. Let $X, Y, Z$ three i.i.d. random variables with distribution $\mathcal{D}$, and $T$ a random variable uniformly distributed in $[...
- 1,677
3
votes
1
answer
395
views
Symmetric distribution optimization problem of distances between points in $[0,1]$
Let $\mathcal{D}$ be a probability distribution with support $[0,1]$. Let $x, y, z$ the outcomes of three i.i.d. random variables $X, Y, Z$ with distribution $\mathcal{D}$, sorted in increasing order, ...
- 1,677
2
votes
2
answers
206
views
non-homogeneous counting process
Consider a counting process $\{N(t), t\geq 0\}$ where the time distribution between any two consecutive events, say $k$ and $k+1$ has a Poisson rate $\lambda(k)$, which is an explicit function of $k$....
- 2,101
2
votes
3
answers
166
views
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
2
votes
0
answers
140
views
Adding elements in a list *in expectation*
Suppose $𝐿_1,…,𝐿_𝑘$ are lists with $n$ elements each. We use a fully independent hash function ℎ to compute a value for each element of each list. (We suppose the hash function returns a value ...
- 43
1
vote
0
answers
38
views
Grid of mostly independent variables
We're given an finite grid of random variables like so:
$$
\begin{bmatrix} A & B &... \\C & \ddots \\ \vdots&&X\\
\end{bmatrix}
$$
a subset of variables on the gird is independent ...
- 309
0
votes
1
answer
159
views
Best bounds on the integral of an increasing function
The following question, somewhat edited here, was asked and then closed at The best bound of the integral of a nondecreasing real function in a closed interval.
Let $F\colon[0,1]\to[0,1]$ be a ...
- 128k
1
vote
1
answer
154
views
If $L_t=\sum_{i=1}^{N_t}Y_i$ is a compound Poisson process, then $\left|\left\{s\in[0,t]:\Delta L_s\in B\right\}\right|=\sum_{i=1}^{N_t}1_B(Y_i)$
Let $H$ be a $\mathbb R$-Hilbert space, $\mu$ be a finite measure on $\mathcal B(H)$ with $\mu(\{0\})=0$ and $(L_t)_{t\ge0}$ be a $H$-valued càdlàg Lévy process on a probability space $(\Omega,\...
CommunityBot
- 1
0
votes
1
answer
3k
views
In linear regression, we have 0 training error if data dimension is high, but are there similar results for other supervised learning problems?
I tried posting this question on Cross Validated (the stack exchange for statistics) but didn't get an answer, so posting here:
Let's consider a supervised learning problem where $\{(x_1,y_1) \dots (...
- 128k
2
votes
2
answers
379
views
Probabilistic combinatorial optimization problem on the distances between pairs of points in $[0,1]$
Let $S$ be a set of $n \gg 1$ points lying on the interval $[0,1]$. Given a point $p\in[0,1]$, let $S_p\subseteq S\times S$ be the set formed by all pairs of points $(x,y)$ with $x,y\in S$, such that ...
- 111
0
votes
1
answer
80
views
Distribution of line segment intersections in random pointsets
let $P$ be a set of $n$ points that are uniformly distributet inside the unit square ore unit circle, and $L=\lbrace\ell_{ij}\rbrace := \lbrace \lbrace \alpha p+ (1-\alpha q)\rbrace\,|\,0\le\alpha\le ...
- 128k
5
votes
2
answers
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