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.
9,023 questions
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Inequality constraints, probability distributions, and integer partitions
I am interested in the possibility of generating probability distributions using inequality constraints. For instance assume that we have three urns with total of a 10 balls. Thus,
$a + b + c = 10$
...
9
votes
4
answers
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Sum of Log Normal random variables [closed]
I would be grateful to anyone who could provide me with some reference concerning the behavior of the sum of Log Normal random variables (need not independent) with respect to a Log Normal random ...
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2
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Random walk is to diffusion as self-avoiding random walk is to ...?
One can view a random walk as a discrete process whose continuous
analog is diffusion.
For example, discretizing the heat diffusion equation
(in both time and space) leads to random walks.
Is there a ...
- 151k
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1
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Does the translation of a random variable preserve its distribution type? [closed]
This might be a very silly question, but I just wanted to make sure I have all the right steps.
Suppose we have a univariate continuous random variable $X$, with some pdf and cdf ${{f}_{X}}(x)$ and $...
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Does positive density imply existence of the density for some part of a decomposition?
Suppose a $\mathcal{H}^{1}$ measurable set $A\subset \mathbb{R}^{n}$ has positive Hausdorff density $\Theta^{1}(\mathcal{H}^{1},A,x)=c>0$ in a point $x\in A$.
If we have a decomposition $A=B\cup ...
9
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2
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674
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Small crown probabilities (and infinite dimensional margin assumption)
My question is:
How do I find sharp upper bounds on $P(|q|\leq \epsilon)$ uniformly over a set of gaussian polynomes $q$ of degree two.
Notations and definitions (to make the question rigorous)
Let ...
2
votes
1
answer
419
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Approximation of the law of a stochastic process
Hello Dear fellows,
I thank you in advance for your help and ideas.
I have just read an article and want you to help me understand the rational behind a part of it.
We have two processes $v_t$ and $...
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2
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912
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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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0
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426
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Maximizing the volume in a family of subsets of a cube
Starting from a question in probability, I arrived to the following optimization problem.
Let $I:=[0, 1],$ and let $A$ be a Lebesgue measurable subset of the $n$-dimensional cube, $A\subset I^n.$ ...
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477
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What is Being Counted? [closed]
There are two urns. One contains five white balls. The other contains
four white balls and one black ball. An urn is selected at random and a ball
in that urn is selected at random and removed. ...
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1
vote
1
answer
257
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Two-Dimensional Gobbling Algorithm
Let (m,n) be an ordered pair of positive integers. While m>0 and n>0, let k_1 be a random positive integer between
1 and m and k_2 a random positive integer between 1 and n. Output (k_1,k_2). Let ...
- 897
8
votes
2
answers
4k
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Infinite sum of random variables: subtle convergence question?
I have a sequence Xj of random variables, each of which individually is uniformly distributed on the unit circle in the complex plane, and a corresponding sequence cj of positive coefficients. My ...
- 12.8k
5
votes
2
answers
3k
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Whitening a random bit sequence
Given an (infinite) stream of uncorrelated random bit with a known "reasonable" bias (say 15-85% 1's) I want to whiten it, e.i. produce a shorter stream of bits that has no bias. The restriction is ...
- 205
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Is there a central limit theorem for bounded non identically distributed random variables?
I have a sequence of centered independent random variables $X_i$ that are all
bounded by one in absolute value. They are not identically distributed, though.
I would like to know if the central limit ...
4
votes
1
answer
232
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Negative Association of Component Size in Random Hypergraph
I have a $d$-uniform hypergraph on $n$ vertices with $k$ hyperedges, where $d << k$ and $n = 4k d^2$ or so.
The hyperedges are placed independently uniformly at random. I would like to have a ...
- 241
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0
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Loynes spaces, also called pseudo-Hilbert spaces
Let me first define my object:
First, a locally convex space $Z$ is called admissible in the sense of Loynes if
$Z$ is complete
There is a closed convex cone in $Z$, called $Z_+$, satisfying (for $x\...
- 2,600
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votes
1
answer
665
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Expectation maximum i.i.d rv´s
If I have a fixed positive integer $N$ and $N$ i.i.d rv´s. $X_1,X_2,...,X_N$, and parameters $a_i$ such that $\displaystyle\sum_{i=1}^N{a_i}=1$, it is well known that there is a global maximum of
$...
26
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3
answers
11k
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L1 distance between gaussian measures
L1 distance between gaussian measures: Definition
Let $P_1$ and $P_0$ be two gaussian measures on $\mathbb{R}^p$ with respective "mean,Variance" $m_1,C_1$ and $m_0,C_0$ (I assume matrices have full ...
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1
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Birkhoff ergodic theorem for dynamical systems driven by a Wiener process
At the risk of asking a stupid question I have the following problem.
Suppose I have a measure preserving dynamical system $(X, \mathcal{F}, \mu, T_s)$, where
$X$ is a set
$\mathcal{F}$ is a sigma-...
- 617
4
votes
3
answers
286
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Medium-Sized Calculations and Organization
This is not a math question as much as a process question. For the first time in my (very short) career, I find myself doing one of those messy calculations, where each 'line' of the calculation can ...
- 63
2
votes
1
answer
1k
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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 $rb$ ...
- 121
5
votes
1
answer
1k
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Concentration of Measure for Power Law
I have a power law distribution $X$ with exponent $c$: $$p(X=t) = \left\\{\begin{array}{cl}(c-1)/t^{c} & t \geq 1 \\\\ 0 & t < 1\end{array}\right.$$
From $X$ I take $n$ independent samples ...
- 241
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2
answers
1k
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Mathematics of the Anthropic Principle [closed]
A form of the anthropic principle is as follows: "We can observe the universe only because we can exist within it in some way such that we can observe it, and it exists such that we can observe it."
...
- 2,383
18
votes
1
answer
2k
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Probability of a point on a unit sphere lying within a cube
Suppose we have a ($n-1$ dimensional) unit sphere centered at the origin: $$ \sum_{i=1}^{n}{x_i}^2 = 1$$
Given some some $d \in [0,1]$, what is the probability that a randomly selected point on the ...
- 1,254
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votes
5
answers
71k
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How do I convert a uniform value in [0,1) to a standard normal (Gaussian) distribution value?
I have uniform value in [0,1). I'd like to transform it into a standard normal distribution value, in a deterministic fashion.
What I'm confused about with the Box-Muller transform is that it takes ...
- 287
3
votes
0
answers
359
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Infinite System of Stochastic Ordinary Differential Equations Coupled by Infinite Graphs
$\ \ \ $ In Time Evolution of Infinite Anharmonic Systems,
Lanford,Lebowitz and Lieb, roughly speaking, proved that for some families of functions $F_v$ $(v\in\mathbb Z^d)$
and a large set of initial ...
- 2,044
10
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1
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527
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Random geometric graphs and spanners
I would grateful to learn of work mixing
random geometric graphs with random graphs under
the
Erdős-Renyi model, and in particular concerning spanners.
Select $n$ points uniformly at random from the ...
- 151k
2
votes
1
answer
100
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Ranking sources at variable(random) frequencies
Hi,
I have this math modeling problem that I need help with. If I have 3 data sources, each being updated at different frequencies, what would be the best way to rank them so the less frequent ...
- 23
6
votes
2
answers
1k
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diameter of a graph with random edge weights
Given a weighted directed graph $G=(V,E, w)$, suppose we generate a new graph $G'=(V,E,w')$ with the same vertices and edges, but now letting the weight of edge $(i,j)$ be
an exponential random ...
- 976
9
votes
1
answer
1k
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Vertex connectivity of random graphs?
Consider simple, undirected Erdős–Rényi graphs $G(n,p)$, where $n$ is the number of vertices and $p$ is the probability for each pair of vertices to form an edge. Many properties of these graphs are ...
- 383
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votes
3
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Analog of Chebyshev's inequality for higher moments
I have a positive random variable $X$ with $E[X] = 1$ and a small number $k$ more moments bounded by constants:
$$E[(X-1)^i] = O(1) \forall i \in \{2, ..., k\}.$$
I'd like to bound the average of $n$...
- 241
4
votes
2
answers
258
views
near independence of markov chain observations at high lags
I have to simulate independent draws from a very complicated distribution. They only feasible way appears to be using MCMC. I was considering running thousands of chains in parallel, but that would ...
- 549
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1
answer
1k
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Points on binary hemispheres of the n-sphere
Let $\mathbb{S}^{n-1}=${$ x\in \mathbb{R}^n| \sum_{k=1}^n x_k^2 =1 $} be the $n-1$ sphere and $n_i\in\mathbb{R}^n$ with components $n_{ij}\in${$-1,1$}$\ \forall\ j=1,2,\dots,n$. There are obviously $2^...
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votes
1
answer
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Probability distribution needed [closed]
Let me clarify my needs. The PDF must comply to:
1. The mean is always in the shorter tail
2. Should have an inverse function
3. Be defined in the interval [0, 1]
4. Should have a shape parameter that ...
20
votes
5
answers
1k
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Iterated Circumcircle
Take three noncollinear points (a,b,c), compute the center of their circumcircle x, and replace a random one of a,b,c with x. Repeat. It seems this process may converge to a point, assuming no ...
- 151k
5
votes
6
answers
2k
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Are all probabilities conditional probabilities? [closed]
We know that $P(A\mid B) = \frac{P(A \cap B)}{P(B)}$. So $P(B) = P(A\mid B)P(A \cap B)$. Thus are all probabilities conditional probabilities? Can one make a probability more accurate by introducing a ...
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1
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A probability exercise related to Central Limit Thm
This exercise appears in K.L.Chung's A Course in Probability Theory, Chapter 7.
Ex.7.1-4
Let ${X_j}$ be independent r.v.'s such that $\max_{1\leqslant j\leqslant n} \frac{|X_j|}{b_n} \to 0$ in
pr. ...
- 41
6
votes
0
answers
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subrandom walkers
Does anyone know of any work on the following model or variants thereof?:
Finitely many chips are distributed on the integers at time 0. To find the distribution at time $t+1$, take all the chips at ...
- 19.7k
12
votes
4
answers
4k
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Mixtures of Gaussian distributions dense in distributions?
It seems that a mixture of Gaussians can approach any probability distribution, as the number of mixture components approaches infinity. Is this true? And if so, is it precise and correct to say ...
- 141
5
votes
4
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629
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Process equivalent to conditional probability
Hi,
Having a random variable $X$ I am trying to find a stochastic process $Z_t$ such that:
$$P[Z_t>T] = P[X > T | X > t]$$
for all $T>t$, or a proof that such a process does not exist.
...
- 667
5
votes
3
answers
945
views
An Integral and derived double integral
Suppose that $f\left(x\right)\geq0$ is continuous on $\left[-\infty,\infty\right]$
and $\int_{-\infty}^{\infty}f\left(x\right)dx=1$. Is it true that
$\int_{-\infty}^{\infty}\left|x\right|f\left(x\...
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votes
1
answer
623
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For which values of $N$ is known the Lieb-Simon Inequality for $Z_N$ Models ?
Background:
Let $\mathbb Z^d$ denote the $d$-dimensional integer lattice with norm $|x|=\sum_i|x_i|$.
For each $x\in\mathbb Z^d$ we associate a spin variable, $\sigma_x$ taking values on the set
$\{...
- 2,044
5
votes
1
answer
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Feynman Kac Formula as appears in Krzysztof Gawedzki's Lectures on conformal field theory
The lecture notes appeared in the second volume of "Quantum Fields and Strings, a course for mathematicians". I would like to understand the derivation of (1.3), the 2-point correlation function:
$$ \...
- 4,466
15
votes
3
answers
2k
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entropy and flatness of densities
I was reading C.R Rao's Linear Statistical inference. Rao presents the entropy of a continuous distribution (expectation of -log density) as a measure of closeness to the uniform distribution, and ...
- 549
8
votes
1
answer
4k
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Skellam distribution: Deep connection between Poisson distributions and Bessel function?
The probability mass function for the Skellam distribution for a count difference $k=n_1-n_2$ from two Poisson-distributed variables with means $\mu_1$ and $\mu_2$ is given by:
$$
f(k;\mu_1,\mu_2)= ...
- 5,935
2
votes
1
answer
467
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Distribution on permutations derived from probability of pairwise orderings
A followup question to Probability estimates for pairwise majority votes - I think it doesn't actually give an answer in any terribly precise sense, but it would give something I'd be happy to use in ...
- 1,331
0
votes
2
answers
429
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E[log(Z_t^2)], proof of convergence with Law of Large Numbers
Hi all,
question:
Let $Z_t$ be an iid sequence with $$\mathbb{E}\log(Z_t^2)<0 $$
Show that $$\sum_{j=0}^\infty Z_t^2 Z_{t-1}^2 ... Z_{t-j}^2 < \infty$$ almost surely
I am supposed to use LLN ...
- 109
4
votes
2
answers
2k
views
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) = ...
- 1,399
1
vote
2
answers
1k
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Sum of digits iterated
Original version.
I believe that it is an elementary question, already discussed somewhere. But I just have no idea of how to start it properly. Take a positive integer $n=n_1$ and compute its sum of ...
- 13.5k
2
votes
0
answers
313
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Finding jump probabilities from mean-occupancy values for positions on a one-dimensional random walk
Please imagine a discrete random walk on a one-dimensional lattice. The lattice consists of a set of $L$ positions, $(x_0, x_1, ..., x_L) \in L$, where $x_0$ is the initial position of the walk (as ...
- 599