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.
9,022 questions
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Do you know of any asymmetric, nonparametric measure of dependence?
A measure of dependence is a way to assign a number (usually normalized between 0 and 1) to a couple of random variable, such that $\delta(X,Y)=0$ if and only of $X$ and $Y$ are independent, and $\...
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Can I relate the L1 norm of a function to its Fourier expansion?
I would like to express the integral of the absolute value of a real-valued function $f$ (over a finite interval) in terms of the Fourier coefficients of $f$. Failing that, I would like to know of any ...
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Sampling from Sine Kernel and Airy Kernel
A determinantal process on the line is a random collection of points on $\mathbb{R}$ such that the probability of $x_1, \dots, x_n$ lying on the random set is $\det (K(x_i, x_j))_{(i,j)}$. Examples ...
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Connectedness of random distance graph on integers
This is not my field, a friend needs the answer for the following question. Suppose we have a decreasing probability function, $p: N \rightarrow [0,1]$ such that $sum_n p(n) = \infty$. Take the graph ...
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Looking for a version of Itô's Lemma
Hi Everyone
I have some difficulties deriving the Stochastic Differential Equation for the following problem, any help or reference would be appreciated.
We are given a Brownian Motion $B_t$ and ...
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When can a freely moving sphere escape from a 'cage' defined by a set of impassible coordinates?
To ask this question in a (hopefully) more direct way:
Please imagine that I take a freely moving ball in 3-space and create a 'cage' around it by defining a set of impassible coordinates, $S_c$ (i.e....
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expected values over binomial distributions
In some works of economics/risk analysis etc., I have seen situations where people take the expected value of a function (such as a utility function/cost function) over a binomial distribution:
$$F(n)...
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"Induced" arrivals in an M/M/1 queue?
I'm a newcomer to the realm of queueing theory, so please bear with me :)
I'd like to model web server traffic with a modified M/M/1 queue.
In the simple case we have two parameters - $\lambda$ for ...
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Spectrum of a generic integral matrix.
My collaborators and I are studying certain rigidity properties of hyperbolic toral automorphisms.
These are given by integral matrices A with determinant 1 and without eigenvalues on the unit circle....
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Estimating the mean of a truncated gaussian curve
Say I have a black box generating data samples, and I want to estimate the parameters of the black box from the samples.
The black box works like this: it has a parameter m (a real number), and to ...
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Formal definition of 'useful' ?
Has anyone worked out a formal, general-enough definition of what is 'useful', so that it could reflectively be used in mathematics? I am aware of the work in utility theory from economics (but ...
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literature on "stratified simulation"
I've thought of an approach to variance reduction that surely can't be new, but I haven't been able to find it published anywhere; I'd appreciate some leads.
Consider some sort of random variable $X$ ...
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Brownian Bridge under observational error
Suppose that $Z_t$ follows a simple discrete random walk $Z_t=Z_{t-1}+e_t$ , where $e_t$ are a bunch of uncorrelated normal variables with arbitrary variance sigma^2, and that there are observations ...
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Polish spaces in probability
Probabilists often work with Polish spaces, though it is not always very clear where this assumption is needed.
Question: What can go wrong when doing probability on non-Polish spaces?
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Entropy of Markov processes
Consider a Markov process $X_t$ with generator $L$ and invariant distribution $\pi$, whose distribution at time $t$ is given by $\pi(t,dx)=\phi(t,x) \pi(dx)$ - in other word, $\phi(t,x)$ is the ...
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Mathematical means of studying large and complex but finite systems?
I want a list of the sort of mathematics/mathematical tools that are applied to the study of complex and probabilistic systems in order to make quantitative and qualitative observations about their ...
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Approximate a probability distribution by moment matching
Suppose we want to approximate a real-valued random variable $X$ by a discrete random variable $Z$ with finitely many atoms. Suppose all moments of $X$ is finite. We want to match the moments of $X$ ...
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Is there an introduction to probability theory from a structuralist/categorical perspective?
The title really is the question, but allow me to explain.
I am a pure mathematician working outside of probability theory, but the concepts and techniques of probability theory (in the sense of ...
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Random generation of subsets using conditional probabilities
Edit: Rewritten with motivation, and hopefully more clarity.
I'm building a site for a card game called dominion. In it, people build 'decks' of 10 distinct cards from a set of (currently) ...
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Is there a theory on two sequences of measures weakly asymptotic to each other?
Suppose $(P_n)_{n\ge 1}$ is a sequence of probability measures on a metric space $E$. Everybody knows what weak convergence of $P_n$ to a measure $P$ is.
Instead, let $(Q_n)_{n\ge 1}$ be another ...
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Problem with a Long Range Correlated Time Series
Consider a stochastic process $X_t$ , $t \in 1,2,3,..,N $.
$X_t$ is a Bernoulli variable and $\Pr(X_t=1) = p$ for all $t$.
The Autocovariance function $\gamma(|s-t|)= E[(X_t - p)(X_s -p)]$ is given
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Conjugate prior of the Dirichlet distribution?
What is the conjugate prior distribution of the Dirichlet distribution?
Edit: Since I asked this question many years ago, I've written a Python library for working with exponential families. Maximum ...
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Book for probability [closed]
I am looking for a good book to study probability. My advisor suggested the "Probability" by Leo Breiman. I am reading it now, it seems rather a dense book, so I would like to ask you guys advice on ...
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Why do Littlewood-Paley projections behave like iid random variables
I have read more than once that the Littlewood-Paley (LP) projections of a function (i.e. decomposing a function into parts with frequency localization in different octaves) behave in some sense like ...
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distribution of degree of minimum polynomial for eigenvalues of random matrix with elements in finite field
This is an attempt to extend the current full fledged random matrix theory to fields of positive characteristics. So here is a possible setup for the problem: Let $A_{n,p}$ be an $n \times n$ matrix ...
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Does central limit theorem hold for general weakly dependent variables?
Say I have $X_{ij}$, $j \le i$ with the property that $X_{ij}$ are centered and identically distributed and $E(X_{ij} X_{ij'}) = o(\exp(-i)))$. Then does $\sum_j X_{ij}$ have Gaussian domain of ...
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Estimate rate of real correct/wrong from 4 answers quiz.
I recently read that one in ten students think the first man on the moon was Buzz Lightyear, a Toy story cartoon. I'm not here to discuss the data in itself, rather, this reading got me into a problem ...
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When is a 1-block factor of a non-Markovian process Markov?
Let $Y$ be a discrete stationary stochastic process. Suppose that $Y$ is not $n$-step Markov for any positive integer $n$. Let $Z$ be a 1-block factor of $Y$. For what condition on $Y$ or the ...
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Stein's method proof of the Berry–Esséen theorem
The relevant paper is "An estimate of the remainder in a combinatorial central limit theorem" by Erwin Bolthausen. I would like to understand the estimate on page three right before the ...
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method of moments and Laplace transform from Shepp and Lloyd
Again from the Shepp and Lloyd paper "ordered cycle lengths in a random permutation", I found this puzzling equality. This one might require access to the paper itself since it's quite a mouthful:
In ...
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Reference request for a "well-known identity" in a paper of Shepp and Lloyd
I ran into a "well-known identity" on page 345 of Shepp and Lloyd's On ordered cycle lengths in a random permutation:
$$\int_x^{\infty} \frac{\exp(-y)}y dy = \int_0^x \frac{1-\exp(-y)}y dy - \log x - \...
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Limit law for the number of local maxima in a square lattice of IID random variables
For $i, j \in \{ 1, \ldots, n \}$, let $X_{i,j}$ be a real-valued random variable uniformly distributed on the interval $[0,1]$. The $X_{i,j}$ are independent.
Let $A_{i,j}$ be the indicator random ...
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Existence of probability measure defined on all subsets
Let $S$ be an uncountable set. Does there exist a probability measure which is defined on all subsets of $S$, with $P({x}) = 0$ for any element $x$ of S ?
If I remove the condition $P({x}) = 0$, then ...
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Constructing Bernoulli random variables with prescribed correlation
For which $n \times n$ correlation matrix $C$ can one construct Bernoulli random variables $(B_1, \ldots, B_n)$ with correlation $C$ ?
Following the approach described in this MO thread, one can ...
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Generalized binomial coefficients and Gaussian density
I ran into an expression calculating the expected value of $\exp(i t \sigma)$ where $\sigma$ is the total number of cycles in a uniformly chosen $S_n$ element. The expression is
$$E_n (\exp(i t \sigma)...
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Is the average first return time of a partitioned ergodic transformation just the number of elements in the partition?
For some reason my thinking is very fuzzy today, so I apologize for the following rather silly question below...
Let $T$ be an ergodic transformation of $(X,\Omega, \mathbb{P})$ and let $X$ be ...
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Motivation for strong law of large numbers
I always find the strong law of large numbers hard to motivate to students, especially non-mathematicians. The weak law (giving convergence in probability) is so much easier to prove; why is it worth ...
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Brownian motion and spheres
Consider a Brownian motion on $[0;1]$. A (finite) discrete approximation of this Brownian motion consists of $N$ iid Gaussian random variables $\Delta W_i$ of variance $\frac{1}{N}$:
$$ W\left(\frac{k}...
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Are there interesting problems involving arbitrarily long time series of small matrices?
Are there well-known or interesting applied problems (especially of the real-time signal processing sort) where arbitrarily long time series of small (say $d \equiv \dim \le 30$ for a nominal bound, ...
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Number of invertible {0,1} real matrices?
This question is inspired from here, where it was asked what possible determinants an $n \times n$ matrix with entries in {0,1} can have over $\mathbb{R}$.
My question is: how many such matrices ...
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Finding a distribution family that is preserved under mixture.
Consider the following
$f_{t+1}(z)=p_{12} f_{t}(z/A)+ p_{21} f_{t}(z/B)+p_{22} f_{t}(z/(A+B))$, where $A$, $B$, and the $p$'s are constants and $f_t$ is a probability distribution. Are there any nice ...
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a unique solution ? iteration involving conditional distributions
consider the following mappings, G and T,
$y(s) = Gx(s)=\exp\left[\sum_{s'}p(s'|s)\log x(s') \right]$
$z(s) = Ty(s)=\sum_{s'}q(s'|s)y(s')e^{-r(s')}$
where $0< x(s)\leq 1$ ,$r(s)<0$ , $s,s'\in ...
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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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Distribution under operations
Let $X$, $Y$, $Z$, and $W$ be i.i.d. copies of a standard gaussian variable, that is in distribution $\mathcal{N}\left(0,1\right)$, then what is the distribution of $\left|\frac{XY}{Z}-W\right|$?
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Likelihood function for sequential random variables
Context
Consider the following sequential data generating process for $Y_1$, $Y_2$, $Y_3$. (By sequential I mean that we generate $Y_1$, $Y_2$, $Y_3$ in sequence.):
$Y_1 = X_1^' \beta + \epsilon_1$
...
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Probabilities and rolling 2 dice
Suppose you start at position 0. You then roll 2 6-sided dice. You move to the integer, call it z, that is the sum of the two dice. You then roll again. If the result of the roll is z', you move ...
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Symmetric groups and Poisson processes
Consider the number of fixed points in a permutation chosen uniformly at random from the symmetric group on $n$ elements - this gives a probability distribution. For $k < n$, the $k$-th moments of ...
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Statistical test comparing two relative frequencies
I'm working with four populations consisting of true/false events. They each have a different mean and variance. I have samples from each, with different sample sizes. Call the percentage of observed ...
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What's the expected number of iterations for this process?
Each step of the process consists of choosing a random integer between 1 and the last number chosen this way. On average, how long does it take to obtain "1" as a result of this process for any given ...
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Google question: In a country in which people only want boys [closed]
Hi all!
Google published recently questions that are asked to candidates on interviews. One of them caused very very hot debates in our company and we're unsure where the truth is. The question is:
...
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