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.
9,024 questions
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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 ...
2
votes
0
answers
560
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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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12
votes
2
answers
997
views
Is there a percolation threshold in the hard discs model?
Take a random configuration of $n$ non-overlapping discs of radius $r$ in the unit square $[0,1]^2$. (You could think of this as taking $n$ points uniform randomly in $[r,1-r]^2$ and then restricting ...
- 7,857
1
vote
1
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242
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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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11
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5
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Brownian motion, martingales, Markov Chains - Rosetta Stone
What are the most
fundamental/useful/interesting ways in
which the concepts of Brownian motion,
martingales and markov chains are
related?
I'm a graduate student doing a crash course in ...
0
votes
3
answers
848
views
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 ...
- 269
3
votes
3
answers
942
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implementations of domino shuffling algorithm
Are there many implementations of the "domino shuffling" algorithm as found in William Jockusch, James Propp amd Peter Shor's Random Domino Tilings and the Arctic Circle Theorem math.CO/...
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2
votes
6
answers
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2D random walk probability to reach a target
Hello Steve,and others thanks.
I was not able to get reference on heat equation which was suggested earlier.
Also the links that was proposed on wike are general and nothing rigours for 2D discrete ...
- 21
8
votes
1
answer
438
views
Potts model simulation
I was wondering what were the state-of-the-art methods to simulate low temperature configurations of Potts-like models that exhibit a discontinuous phase transition. For models with a continuous phase ...
- 2,133
62
votes
7
answers
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Why is the Gaussian so pervasive in mathematics?
This is a heuristic question that I think was once asked by Serge Lang. The gaussian: $e^{-x^2}$ appears as the fixed point to the Fourier transform, in the punchline to the central limit theorem, as ...
- 631
21
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11
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What are some good examples of non-monotone graph properties?
It seems that many, if not almost all, of the properties studied in graph theory are monotone. (Property means it is invariant under permutation of vertices, and monotone means that the property is ...
- 7,857
3
votes
1
answer
472
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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 ...
- 995
1
vote
1
answer
152
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References for Poisson and Lexis trials
I have been trying to find more information on Poisson and Lexis trials (generalizations of Bernoulli trials), but I have failed to find anything outside of MathWorld (I went through a number of ...
- 203
3
votes
1
answer
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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 ...
- 453
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3
answers
2k
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Wiener process related counterexample
The Wiener process is defined by the three properties:
1. $W(0) = 0$,
2. $W(t)$ is almost surely continuous, and
3. $W(t)$ has independent increments with $W(t) - W(s) \sim N(0, t-s)$ (for $0 ≤ s &...
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votes
3
answers
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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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6
votes
1
answer
453
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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 ...
- 8,512
3
votes
7
answers
4k
views
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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35
votes
1
answer
2k
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Random walk inside a random walk inside...
Let $G=(V,E)$ be a graph and consider a random walk on it. Let $G'=(V',E')$ be a subgraph consisting of the vertices and edges that are visited by the random walk.
Question 0: Is there a standard ...
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16
votes
3
answers
2k
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A random walk on random lines
I am wondering if this random walk remains finite with positive probability.
Start with three lines $A,B,C$ that are extensions of an equilateral triangle.
Let $p_0$ be one corner. Generate a line $...
- 151k
7
votes
0
answers
717
views
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_{...
- 4,304
3
votes
1
answer
693
views
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. ...
- 133
3
votes
2
answers
919
views
a point process is characterized by its void probabilities
Consider a planar point process $X$ and call $N_A = \text{Card}\big( X \cap A\big)$ the number of points inside the subset $A \subset \mathbb{R}^2$. If one knows the law of $(N_{A_1}, \ldots, N_{A_r})$...
- 2,133
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votes
3
answers
1k
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Imaginary exponential functional of Brownian motion
Thanks to the work by M. Yor and colleagues, much is known about the following exponential of Brownian motion:
$X= \int_0^{\infty}{\rm d}t \ e^{-t + g \ B(t)}$
where $g$ is a real scale parameter.
...
- 141
1
vote
2
answers
4k
views
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 ...
- 137
-1
votes
1
answer
247
views
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) ...
- 11
11
votes
2
answers
2k
views
How many non-equivalent sections of a regular 7-simplex?
Suppose we have a regular 7-simplex in $\mathbb{R}^8$ defined by vertices <1,0,0,...,0>, <0,1,0,..,0>,...,<0,...,0,1>. A section is a 3-dimensional linear subspace of $\mathbb{R}^8$ that ...
2
votes
2
answers
625
views
x-th moment method
For a real-valued random variable, $X$, the first moment method, is simply
$$P(X\ge\mathbb{E}[X])>0.$$
$\DeclareMathOperator\Var{Var}$This can be extended to the second moment quite easily:
$$\...
- 316
21
votes
1
answer
3k
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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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28
votes
2
answers
2k
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"Are you taller than the average of those who are taller than the average?"
I've met tall people. That is: people taller than the average. Every now and then we encounter really tall people, even taller than the average of tall people i.e. taller than the average of those who ...
- 23.3k
9
votes
5
answers
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estimate the error term in CLT
Let $X_m = \frac{1}{\sqrt{m}}\sum_{k=1}^m Z_k$ where $Z_k$ are iid equally likely on $\{\pm 1\}$. Then $X_m$ convergens to $X \sim \mathcal{N}(0,1)$ in distribution by CLT.
Let $f$ be a smooth ...
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91
votes
8
answers
16k
views
Is there a natural random process that is rigorously known to produce Zipf's law?
Zipf's law is the empirical observation that in many real-life populations of $n$ objects, the $k^\text{th}$ largest object has size proportional to $1/k$, at least for $k$ significantly smaller than $...
- 114k
3
votes
5
answers
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Recommended book for introduction to Chaotic dynamics? (application in probability distributions)
I'm just starting some research and I need a good introductory book in the topic of chaotic dynamics. Does anyone have a suggestion? Thanks.
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7
votes
1
answer
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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 ...
- 71
0
votes
1
answer
2k
views
kalman filter: understanding the mathematical part
i am currently reading the Probabilistic robotics book where the filters are discussed.
Such filters as kalman filter or particle filters.
Now I can understand one thing while reading about the ...
3
votes
1
answer
578
views
Why doesn't Stein effect happen for multinomial distributions?
(Medeen, et all, 1998)" show that Maximum Likelihood estimate is admissible for multinomial distribution under squared error. On other hand, James and Stein showed that arithmetic average is not an ...
- 2,252
8
votes
3
answers
847
views
Random linear recurrence relations
Problem
I am interested in the random recurrence relation of the form $x_{n+1}=\alpha x_n \pm \beta x_{n-1}$ where $\alpha$, $\beta$ are known constants and the $\pm$ sign is chosen with equal ...
- 3,217
24
votes
3
answers
4k
views
What upper bounds are known for the diameter of the minimum spanning tree of $n$ uniformly random points in $[0,1]^2$?
Let $P$ be a pointset consisting of $n$ uniformly random elements of $[0,1]^2$. It is known that the diameter (greatest number of edges in any shortest path between two points) of the Delaunay ...
- 4,922
5
votes
2
answers
2k
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Process for a Gamma distribution with non integer shape parameter
I am sampling the distribution of lifetimes of computers participating in massive volunteer computing initiatives (BOINC projects). While a phenomenological Weibull distribution makes a good ...
- 437
3
votes
5
answers
986
views
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, \...
- 144
2
votes
5
answers
4k
views
Random walk on a two-dimensional uniform grid
Hi,
consider the following random walk on the lattice $\{0,\dots,n\}^2$. It starts at $(0,0)$ and then move either up or right, with probability respectively $p$ and $1-p$. Once it reaches the right ...
- 23
0
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0
answers
138
views
Why do I not use post hoc tests with a 2 x 2 factorial?
I know this is an obvious answer. I am probably over thinking what I'm doing, but I cannot recall. Does it have to do with not having enough variables to compare the various means?
5
votes
0
answers
308
views
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 ...
- 24.8k
13
votes
1
answer
654
views
Is the nearest walk to Brownian motion uniform?
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 continuous functions $f:[0,1]\rightarrow\mathbb R$ ...
- 24.8k
2
votes
0
answers
351
views
Distribution of transformed multinomial variable?
Suppose we have a uniform multinomial distribution over $2^d$ outcomes. Multinomial coefficients give distribution of vector valued variable $v$ where $v$ is the vector of observed counts.
Is there a ...
- 2,252
14
votes
0
answers
587
views
Why, and how badly, does the proof of "no percolation at the critical point in half-spaces" fail for full spaces?
The proof by Barsky et. al. that there is no percolation in half-spaces proceeds by a dynamic renormalization argument. The proof couples critical percolation in the half-space $\mathbb{H}^d$ with a ...
- 4,922
4
votes
2
answers
2k
views
Elo Rating System Help with the Maths around number of matches
I'm creating a system that will allow people to rate images.
My idea is to use an Elo Rating system (http://en.wikipedia.org/wiki/Elo_rating_system) for each image and then use crowdsourcing to have ...
- 151
1
vote
2
answers
661
views
Markov chain convergence problem.
Consider a markov chain matrix P of size n x n (n states).
P is known to be:
1- there are at least two absorbent states. one of them is denoted by null. (thus, we have that P_null,null = 1)
2- For ...
- 27
7
votes
3
answers
966
views
Expectation of a simple function of multivariate gaussians iid rvs
I would like to compute analytically the following expected value:
$$ E\left( \frac{X_i^2}{\sum_j \lambda_j^2 X_j^2}\right) $$
where the $X_i \approx N(0,1)$ are iid.
It seems to be an elementary ...
- 461
-2
votes
3
answers
2k
views
Convergence of a markov matrix
Consider a markov chain matrix P of size n x n (n states).
P is known to be:
1- Not irreducible (i.e. there exist at least a pair of states i, j such that we cannot go from i to j)
2- Not all ...
- 27