Questions tagged [pr.probability]
Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
9,027 questions
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Haar measure on $O(n)$ reduced to simpler probability space
The background of this question is how a random variable $X$ on the orthogonal group $O(n)$ whose distribution is the normalized Haar measure $\mu$, i.e., $\mu( O(n) ) = 1$, can be realized on a ...
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probability P that all circles are connected with each other.
Let N circles with homogeneous radius r are deployed with Poisson distribution in area A. These circles are connected if there euclidean distance is less than r.what is the probability P that all ...
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Giving a general term of a recursive function, and upper bound for it
Let a constant $B \ge 1$, and let $l_1 = 0$, $b_1 = 0$ be the values of $l$ and $b$ (respectively) at time $t = 1$.
Let $l_{t+1} = l_t + 1$ if $b_i < B$, and $l_{t+1} = l_t$ otherwise
Let $b_{t+1}...
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Removing edges from Erdős–Rényi graph to make two nodes disconnected
Consider a Erdős–Rényi graph on $n$ nodes, say $1,2,\ldots,n$, ($n\geq 3$) such that the probability of edge between any two nodes is $c/n$. I wish to know if there is a result that says
"There ...
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Independence using reflecting brownian motion
Suppose $X$ and $Y$ are two Brownian motions such that $|X|$ and $|Y|$ are independent. Then it is easy to show that $\langle X,Y \rangle =0$ using the Tanaka formula, for example, and thus $X$ and $Y$...
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Concentration of sums of random vectors under moment conditions on the marginals
Let $X_1,\dots,X_n\in {\bf R}^d$ be $d$-dimensional iid zero mean random vectors with covariance matrix $\Sigma$. I am interested in tail bounds for the Euclidean norm
$$N_n\equiv \frac{1}{\sqrt{n}}\|...
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The balls and bins model: bounding the marginal contributions in the m>>n regime
Consider the standard balls and bins process, where $m$ balls are thrown into $n$ bins, and consider the case where $m >> n$. Denote the load on bin $i$ by the RV $L_i$.
Given a set $S \...
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When is the support of a Radon measure separable?
Let $X$ be a topological space, equipped with its Borel $\sigma$-algebra $\mathcal B(X)$, and let $\mathbb P$ be a Radon probability measure on $(X, \mathcal B(X))$. Recall that the support of the ...
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Elliptic Harnack inequality for 1D Schrodinger operator?
For a nonnegative polynomial $V: \mathbb{R} \to \mathbb{R}$, write $H = -\Delta + V$. I am wondering if there is an elliptic Harnack inequality for H. That is:
There exist $C_{H} > 0$ and $\delta \...
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Quasicompactness of transfer operators associated to IID matrix products
Let $P^1$ denote one-dimensional real projective space, and for each $A \in GL(2,\mathbb{R})$ let $\overline{A}$ denote the homeomorphism of $P^1$ induced by $A$. I am currently reading a paper which ...
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Asymptotic independence in a multinomial setting.
Let $(X_1,\ldots,X_r)$ be a multinomial vector with parameters $n$ and $1/r$, i.e., we throw $n$ balls into $r$ bins, with a uniform probability for each ball to land in each of the bins. As is well ...
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Is there any finitely-long sequence of digits which is not found in the digits of pi? [closed]
I know it's likely that, given a finite sequence of digits, one can find that sequence in the digits of pi, but is there a proof that this is possible for all finite sequences? Or is it just very ...
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The Lindeberg Condition
Suppose $Z_{1}, Z_{2}, \dots, $ are independent and identically distributed random variables with mean 0 and variance 1. Put $X_{nk}=\sigma_{nk} Z_{k}$ for $n=1,2, \dots$ and $k=1, 2, \dots, r_{n}$, ...
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Metrized categories
Motivation: Let $\Gamma = (V,E)$ be a directed graph. To each edge $e \in E$, choose a value $\kappa^e \in \mathbb R$, representing the cost of transporting one unit of "stuff" through the edge. Let $\...
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Conditional law of an Ito's stochastic integral
Consider $B=(B_t)_{t\geq 0}$ real $\mathcal F_t$ - brownian motion starting at zero, in a probability space $(\Omega, \mathcal F, (\mathcal F_t)_{t\geq 0}, \mathbb P)$. Then, consider a new real $\...
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The fraction of the sphere a fixed distance from a subspace
The following problem has a beautiful geometric interpretation in terms of the proportion of points on the Euclidean sphere in $\mathbb{R}^d$ that lie at least a certain distance away from a $k$-...
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n balls, k colors, expected color change problem [closed]
I was asked this question during my interview recently and despite the amount of thinking i put into this, I am yet to figure it out:
Given $n$ balls which are painted by $k$ colors. Let $s_i$ number ...
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derivative in the Wasserstein space
Villani gives the following formula to find the gradient of a function $F$ of a probability density function $\rho$ in the Wasserstein space :
$$\nabla_W F(\rho) = -\nabla.(\rho \nabla \frac{\delta F}{...
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Distribution for probability of an incorrect inference based on a comparison of only two samples?
I'm trying to demonstrate the problems of how taking a sample and assuming it reflects the population accurately can be problematic.
Imagine say an urn with some large number of balls, black and ...
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A machine learning application question
I am familiar with basic probabilities, random processes but not so much of machine learning methods. This is the problem I am trying to solve.
I want to predict the nature of user activity on a ...
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Probability $k$ bins are non-empty.
The following problem arises in the analysis of Bloom filters.
Consider $m$ bins and $N=nk$ balls placed uniformly at random into the bins. A query chooses $k$ bins uniformly at random and asks if ...
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How fast is discrete-time diffusion on a continuous set?
This question is inspired by Joseph O'Rourke's beautiful answer to my previous question.
Let $\mathbb{S}^{d\times n}$ denote the set of real $d\times n$ matrices whose columns have unit norm and sum ...
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Another colored balls puzzle (part II)
The same colleague as in Another colored balls puzzle asked me the following variant which she called "part II".
Imagine you have $n$ balls in a bag that are colored from $1$ to $n$. At each turn ...
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Finding conditions on unspecified CDF that permit a solution to an equation
[A duplicate thread can also be found at
https://stats.stackexchange.com/questions/59450/finding-conditions-on-unspecified-cdf-that-permit-a-solution-to-an-equation ]
Let $F(\alpha) := \mathbb{P}(\...
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Joint distribution from multiple marginals
Consider an experiment consisting of a repeated trial with two random Bernoulli (=binary) variables, A and B. Each trial consists of multiple outcomes for both A and B. Each trial has the same number ...
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Probability that one RV will exceed many others
Assume the $1 \times N$ vector
$\mathbf X = [X_1, X_2, \ldots , X_N]$
contains i.i.d. normal samples such that $\mathbf X$ has a multivariate normal distribution. Now assume another random variable $...
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Blue and red balls puzzle
I was sent this puzzle and wondered if it is known or if its origin is known? (I see colored ball puzzles are also in vogue.)
Consider a bag with $n$ red balls and $n$ blue balls. At each turn you ...
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An interesting version of the problem “balls into bins”
Consider n people, each has k identical balls. Each people choose k different bins from m bins, constrained by the condition that there are no two people choose exactly the same k bins. For instance, ...
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Continuous dependence of the expectation of a r.v. on the probability measure
$\newcommand{\bsV}{\boldsymbol{V}}$ $\newcommand{\bsE}{\boldsymbol{E}}$ $\newcommand{\bR}{\mathbb{R}}$ Suppose that $\bsV$ is an $N$-dimensional real Euclidean space. Denote by $\newcommand{\eA}{\...
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Generalization of hypergeometric distribution?
Consider a Polya urn model in which we have $N$ balls of which $k$ are black and the rest are white. We repeatedly draw balls (without replacement) in "batches" of size $m$ (assume $m$ divides $N$), ...
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Savings property: A transformation which turns nonnegative martingales into uniformly integrable ones
Background
I work in a subfield of computability theory called algorithmic randomness. We have been using martingales as long as probability theory (going back to work of von Mises). However, since ...
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probability calculation
Given $m\cdot e$ balls, $b$ of which are black (suppose the rest are white balls). Randomly put the balls into $m$ baskets, with $e$ balls in each basket. What is the probability of the event that ...
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Computing hypergeometric function of matrix argument
In the context of the Bingham probability distribution the ${ }_1F_1$ hypergeometric function of matrix argument naturally arises as a normalization constant of the probability distribution function. ...
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Probability distribution for two-state system that depends on residence time
I am a statistical physicist, and I've come across a problem that I don't know how to solve. I believe my issue lies with how to formulate it mathematically. I'd be very grateful for any assistance, ...
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Transition time in finite voter model
I believe the following problem is related to something called the "voter model" in statistics. This is not my area of expertise so please forgive me if the answers turn out to be well known.
...
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Expected minimum Hamming distance with overlaps
Let's say we sample two random binary vectors, one called $A$ of length $n$ and the second called $B$ of infinite length. Now we compute $X_k= \min_{i\in[k]} w(A \oplus B[i,i+n-1])$ where $w$ computes ...
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Random graphs nonisomorphic to unit distance graphs
I've encountered an interesting problem but can solve it only partially:
Prove that random graph $G\sim G\left(n,\frac cn\right)$, $c=const$, almost surely is isomorphic to some unit distance graph ...
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Is every submartingale a convex function of a martingale?
Is every submartingale a convex function of a martingale?
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Pruning copies of an element from a multiset via a uniform random selection process - does vigilance matter?
This is an extension of a previous question of mine (nicely answered by Douglas Zare): Filling a bin with one type of element when uniformly selecting from a set of two (with bias)
Say I fill a ...
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Filling a bin with one type of element when uniformly selecting from a set of two (with bias)
I have two bags: one filled with red marbles, and one filled with blue marbles. I would like to fill a bin with only $k$ red marbles and no blue marbles. However, I can only sample (with replacement)...
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Shortest loop containing 0 in continuum percolation
I am interersted in continuum percolation with intensity $\lambda>0$. Formally, let $X$ be a Poisson point process in $\mathbb{R}^d$ with intensity $\lambda$ and $G$ the graph obtained by ...
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Distributions of Time Derivatives of Stochastic Processes
Suppose $u(t,\omega), t \geq 0$ is a continuous time stochastic process with smooth paths so $\frac{d}{dt} u(t,\omega)$ exists for all $\omega$. Suppose you know the distribution of $\frac{d}{dt} u(t,...
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Optimizing a stochastic "flip and prune" procedure for selecting a subset of coins
I place some number of coins, $(c_1, ..., c_N) \in C$ on a table, where each coin is originally tails up. Let's call the "tails" state $0$ and the "heads" state $1$. I then perform the following ...
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Probability of random (0,1) Toeplitz matrix being invertible
A Toeplitz matrix or diagonal-constant matrix is a matrix in which each descending diagonal from left to right is constant.
What is the probability that a random $n \times n$ binary Toeplitz ...
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Fundamental inequality of entropy in random walks
I'm looking for a reference for an inequality related to the "fundamental inequality" about entropy and rate of escape of random walks (on the Cayley graph of a group). Namely,
$\textbf{Question}$: ...
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Question on two measures of correlation
For two $\sigma$-fields, $\mathcal{A}$ and $\mathcal{B},$ we have the notion of HGR maximal correlation
$$\rho(\mathcal{A},\mathcal{B}) = \sup \frac{Efg-Ef.Eg}{\sqrt{\mathsf{Var}(f).\mathsf{Var}(g)}}...
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Bounding statistical distance by matching moments
Suppose we have distributions $p(x)$ and $q(x)$ both supported on integers in $[-n, +n]$. We want $p$ and $q$ to have statistical (total variational) distance of at most $\epsilon$.
Is there a ...
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Probabilities of a random walk exiting a set
Let $F$ be a finite connected set in a graph (soon to be the Cayley graph of a group) and $\mathrm{Ex}_x^F$ be the function on the vertices in $F^c$ which are neighbour to vertices in $F$ defined as ...
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Probability of difference between elements in a sorted set
Given a set of naturals [n]:{1,2,3,...,n}, we repeatably select m elements from [n] to make a sorted set ${x_1,x_2,...,x_m}$ satisfying $x_i >= x_j$ for i>=j.
What the probability $p(i<=k)$ ...
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Generating Random Young Tableaux: A peculiar probability identity
In the paper by Greene, Nijenhuis and Wilf, an algorithm is proposed for generating uniformly random Young tableaux of shape $\lambda$. The algorithm is to uniformly randomly pick a starting cell, and ...
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