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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What are Central Limit Theorems and why are they called so?
I know two opinions:
1) "Central" means "very important" (as it was central problem in probability for many decades), and CLT is a statement about Gaussian limit distribution. If the limit ...
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"continuous" and "discontinuous" phase transitions in branching processes.
Consider a Galton-Watson branching process, with offspring distribution
$\mathbf{p}=(p_0, p_1, \dots, p_n, \dots)$.
Let $O$ be the root of the branching process.
Write $\eta=P(\text{process survives ...
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Expectation of first positive value in random walk
Let $p$ be a parameter in $]0,1[$. Let $(X_k)_{k\geq 0}$ be an independent, identically distributed sequence of random variables, such that each $X_k$ takes values only in
$\lbrace -1, \frac{1-p}{p} \...
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Coordinatizing the disk via Brownian motion
Divide the unit circle into three arcs, and let $z$ be a point in the open unit disk. Is there a simple formula for the probability that Brownian motion started at $z$ will hit one particular arc ...
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When is it possible to construct a joint law from its two-dimensional marginals?
My question is much more specific than the title:
Given a symmetric
distribution $\Xi$ on $\mathbb R^2$, when is it possible to construct a sequence $\xi_1,\xi_2,\dots$ of random variables such ...
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The conditions in the definition of Poisson process (and a Lévy process generalization)
Last week, George Lowther provided a rather sophisticated counter-example of a continuous process $\{W(t):t \geq 0\}$ with $W(0)=0$ and $W(t)-W(s) \sim {\rm N}(0,t-s)$ for all $0 \leq s < t$, yet ...
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Quantum probability experiment?
I am looking for an example (or definition) of a quantum probability experiment (if there is such a thing). Ideally it should have these properties:
Be purely mathematical; no mention of physics or ...
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Numeric problem when evaluating log of a pdf
In maximum likelihood estimation, one typically needs to compute the log (natural log) of probability values. When a probability, say $p(x)$, becomes so close to zero, $log(p(x))$ returns -Inf. What ...
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How to calculate the probability of N normal variable being in increasing order?
Suppose we have $n$ normal variable $X_1,X_2,\dots,X_n$, with corresponding mean $\mu_1,\dots,\mu_n$ and sd $\sigma_1,\dots,\sigma_n$. What is the probability of $X_1 < X_2 < \dots < X_n$, i....
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Probability space analogue of Cauchy-Schwarz inequality
Suppose we have two sets of discrete events, $A$ and $B$. Then I think it is true that:
$$2\sum_{i \in A, j \in B}\Pr[i\ \textrm{AND}\ j] \leq \sum_{i \in A}\Pr[i]+ \sum_{j \in B}\Pr[j] +\sum_{i, j \...
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Expectation of RVs with Poisson-type decay
I need to bound the expectation of a nonnegative random variable that satisfies a Poisson-type tail bound:
$\mathbb{P}( X \geq t ) \leq \min( d \cdot (\frac{a}{t} )^{t}, \ 1)$ for $t > 0$
where $...
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Are there nonequivalent randomnesses?
There are nonequivalent geometries, nonequivalent groups finite and infinite, nonequivalent logics ( fregean and nofregean http://www.formalontology.it/suszkor.htm), even nonequivalent logicians;-)
...
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is there an interpretation to the inverse of $I-M$ in multitype branching process, where $M$ is the mean matrix?
Assume we have a multitype branching process, i.e., we have a mean matrix $M_{ij}$ and $M_{ij}$ is the expected count of generating $j$ from $i$ in one time step, i.e.:
$M_{ij} = \sum_{r} n(r,j)P(r | ...
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Heaviside Step Function of a Random Variable
I have a random variable $X$ and I want to find the probability density function from transforming it through the Heaviside step function. So
$Y = H(X)$
where the $H$ is the Heaviside step function ...
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State of the Art in Stable limits, embeddings, etc
This is a fairly broad request for references. I've tried a few hours of googling, but the usual process of chasing names and references doesn't seem to be converging on any must-read books or ...
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An eventually different function adding no Solovay real nor dominating function?
Definitions
I believe set theorists have studied all of the following three notions in the context of forcing extensions of a model of ZFC, $M$ (hopefully the terminology is the standard one).
A ...
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What is this probability distribution?
Suppose we have a family $F_0,F_1,\dots$ of independent random variables which take the value $1$ with probability $p$ and $0$ otherwise; let $\delta$ be a number between $0$ and $1$. Let
$X_n = \...
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Statistics for Second order properties of Random graphs
Hi!
Let G(N) be the number of graphs with vertices {1, 2, ..., N} and GN(F) be the number of those of them which satisfy graph property F. There is a beautiful result by Glebskii and Fagin that limit ...
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For Ising models on finite graphs, is the gradient of Z (w/r/t coupling and field) easier to compute than Z?
Suppose we have a graph $G$ with $n$ vertices, edgeset $E$, $\mathcal{X}=\{1,-1\}^n$. The partition function of the spin-1/2 Ising model on $G$ is
$$Z(J,h)=\sum_{x\in \mathcal{X}} \exp\left(J \sum_{(...
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Probability of system failure in a distributed network
I am trying to build a mathematical model of the availability of a file in a distributed file-system. The system works like this: a node $x$ stores a file $f$ (encoed using erasure codes) at $rb$ ...
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Tightness of probabilty distributions
Let $\mathcal{P}(\mathbb{N})$ be the set of all probability mass functions on $\mathbb{N}=\{1,2,\dots \}$. Let $E$ be a closed(with respect to pointwise convergence, or equivalently the total ...
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In an inductive family of groups, does the probability that a particular word is satisfied converge?
We have some group word $w$ in $k$ letters. We say a $k$-tuple of group elements $\vec{g} = (g_1, g_2, \ldots , g_k) \in G^k$ satisfies the word $w$ if $w$ gives the identity at $\vec{g}$. More ...
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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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You pass X people and Y people pass you: how relatively fast are you?
This question occurs to me every time I go jogging. I suspect every runner probabilist in the world must have thought of it (though I'm no probabilist), but I could not specifically find it online. I ...
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extensions to Bernstein's inequality I could use for bounding probability of a union?
I have a set of dependent Bernoulli variables $X_i$ for $i \le N$, with probability $\epsilon$ for the event $X_i=1$.
I want to bound the probability that $\sup_i X_i \ge 1$, i.e., I want to know ...
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Path cardinality for random $(a+b)$-ary infinite trees
Consider a random infinite binary tree $T(a,b)$, so that $a$ denotes the probability of a left edge branching from any root-connected node,and $b$ denotes the probability of a right edge branching ...
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how to formalize a notion of symmetric set difference probability? [closed]
I saw in a paper an argument, that seems simple. (Let $\triangle$ be the triangle operator of symmetric set difference between two sets)
It states that if $P(A \triangle B) < \epsilon$, for some ...
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Generalization of repeated error function integral
Is there a name for the following integral?
$f(x, y, n) = \int_y^\infty (t - x)^n \exp(-t^2) dt$
The parameter $n$ is positive. The first priority is integer $n$, but more generally the case of real-...
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The question about Kolmogorov tightness criterion
We know about Kolmogorov Criterion for the tightness of a stochastic process $X_n(t)$
1.The sequence $(X_{n}(0))_{n\geq0}$ is tight.
2.There exist constants $\gamma\geq0$,$\alpha>1$, $K>0$ and ...
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Is the nearest walk to Brownian motion approximately uniform?
This is a follow-up to an earlier MO question.
Let $W:[0,1]\rightarrow\mathbb R$ be standard Brownian motion with $W(0)=0$.
Let $F_n$ denote the collection of all the $2^n$ many piecewise linear ...
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What the the probability distribution of a mean?
There is an unknown set of values of unknown size, from which a known subset of N values is drawn at random.
Based on the known random subset, what is the probability distribution of the mean of the ...
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Measuring the randomness in random numbers
I'm looking to write a program to investigate a few random number algorithms. Basically I am looking to see if the spread of numbers is indeed randomly distributed enough. What kind of statistical ...
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Density function for a multivariate Bernoulli-like distribution
I'm looking for a distribution to model a vector of $k$ binary random variables, $X_1, \ldots, X_k$. Suppose I have observed that $\sum_i X_i = n$. In this case I do not want to treat them as ...
- 24.8k
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Path continuity for (closed) martingales?
Take a time interval $[0,T]$, and a filtered probability space $(\Omega,P,\mathcal{F},\mathcal{F}_t)$. If $X \in L^1(\mathcal{F}_T)$, then $M_t = E [X \ | \ \mathcal{F}_t]$ is a martingale. If I ...
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Examples where Kolmogorov's zero-one law gives probability 0 or 1 but hard to determine which?
Inspired by this question, I was curious about a comment in this article:
In many situations, it can be easy to
apply Kolmogorov's zero-one law to
show that some event has probability 0
or 1, ...
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Is the truncated Brownian motion of the class DL?
Let $W$ be a standard Brownian motion under given probability space.
For a given constant $a$, $W^a$ is a truncated Brownian motion by stopping time
$T^a = \inf(t>0:W(t) = a)$. That is, $W^a(t) = ...
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Change of Time in Stochastic Integral
Hi everyone,
Let's be given $I(0,t)$ a Stochastic Integral with respect to a local martingale $ M_t$ of the form :
$I(0,t)=\int_0^t h(s_-)dM_s$ with $h\in L(M)$ (for example $h$ is an adapted ...
- 15.4k
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initial condition of a diffusion approximation
I am trying to prove that a certain sequence of Markov chains $x^N_k$ converges towards a diffusion process. The invariant measure of $x^N$ is $\pi^N$ and the Markov chain $x^N$ is started in ...
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Two-dimensional random walk
Suppose we have a particle in the plane at the origin $(0,0)$. It moves randomly on the integer lattice $Z^2$ to any of the adjacent vertexs with equal probability $1/4$. What's the probability of ...
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Numerical Solution to Inverse Integral (Pseudo Random Number Generation)
If I have an arbitrary positive monotonically decreasing function $f(x), x \in [0,\infty]$, is there an 'efficient' method for finding $y$ in:
$r = \int\limits_0^y f(x) dx $
for a known $r \in [0, \...
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How to factorize the joint probability of an arbitrary graph whose nodes are random variables?
This question is about graphical modeling of joint probability functions, Markovian property and Markov random fields.
Suppose we have an undirected graph G where each node represents a random ...
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The typical size of a random element in a Banach space
Let $X$ be a separable Banach space, and let $\mathbb P$ be a Radon probability measure on $X$ with zero mean and covariance operator $K : X^* \to X$. Let $x$ be an $X$-valued random variable with ...
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How to tell if two random polynomials are identical
Let t be a positive real number. Let P(x) and Q(x) be two random polynomials with integer coefficients. If P(t) = Q(t), then what is the probability that P(x) is not identical to Q(x)?
Will it make a ...
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Inequality involving probability measures [closed]
I have been working on a problem(alternate minimization) where I want to establish an inequality in which I am stuck.
An $\alpha$- parameterized version of the divergence(Kullback-Leibler) takes the ...
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Is there a continuous-time version of Kingman's subadditive decomposition theorem?
Kingman's subadditive ergodic theorem (see this article) states that if $x_{m,n}$ is a real valued process indexed on the set of pairs of non-negative integers $m < n$ satisfying:
$x_{l,n} \le x_{...
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Sequence of p draws without replacement with biased probabilities
Hi
I have a problem which i find hard to modelize.
Suppose i have an urn with $N$ marbles. Among these marbles, one is white and all the other ones are black. I draw $P$ marbles without replacement. ...
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Kalman filtering: 1D case
How will the kalman filtering model look like in the case when I just receive some data and want to filter them from noise? The data is actually an acceleration of some object.
So the system must be ...
CommunityBot
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Coalescing random walks: a bound for the full coalescence time?
Start a random walk from each vertex of a graph $G$. Let the walkers evolve independently, except that when two of the walkers meet (ie. occupy the same vertex at the same time), they coalesce into ...
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Intuitive Proof of Cramer's Decomposition Theorem
Cramer's decomposition theorem states that if $X$ and $Y$ are independent real random variables and $X+Y$ has normal distribution, then both $X$ and $Y$ are normally distributed. I've seen a few ...
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Concentration results for non-standard Gaussian random vectors.
Given a $c$-Lipschitz function $f(X):\mathbb{R}^d \rightarrow \mathbb{R}$, and given that $X \in \mathbb{R}^d$ is a Gausssian random vector centered at $\mathbb{w} \in \mathbb{R}^d$ (not at zero) ...
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