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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Conditional probability, deviation from the uniform distribution
Let $N\in\mathbb{N}$ and $G$ the group $\mathbb{Z}/n\mathbb{Z}$.
Let $q< N$ and:
$a_1, ..., a_q$ pairwise distinct
elements of $G$
$b_1, ..., b_q$ pairwise distinct
elements of $G$
$x_1, ..., x_q$...
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Intersection of an uncountable number of sets.
Let $\mathcal{I}$ be an uncountable set. Let $(\Omega, \mathcal{F},\mathbb{P})$ be a probability space, and $E_i, i\in \mathcal{I}$ be a measurable set such that $\mathbb{P}(E_i)=1$. What can we say ...
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What distribution(s) of delays make(s) timing attacks hardest?
$H$ is (Shannon) entropy.
In terms of the positive real number $t$, what distribution(s) $\hspace{.01 in}X$ on $\:[0\hspace{.005 in},\hspace{-0.03 in}\scriptsize+\normalsize\infty\hspace{-0.02 in})$...
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Brownian motion above another one.
We define
$p_T(f):= \mathbb{P}(\forall_{s\leq T}B_s \geq f(s)-1),$
where $B$ is a Brownian motion such that $B_0 =0$ and $f:\mathbb{R}_+ \mapsto \mathbb{R}$ is some (continuous) function. I am ...
- 1,491
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Probability that a "closable" self-avoiding random walk forms a polygon
Consider a self-avoiding random walk on an infinite graph (for concreteness, the grid of 2-dimensional lattice points $\mathbb{Z}^2$), in which on each step, the next position is chosen uniformly at ...
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Sampling without replacement: probability for total successes from successes in sample?
Consider drawing $n$ balls from an urn containing $N$ balls, of which $m$ are red. If i know $N$, $m$ and $n$ i can use the hypergeometric distribution to calculate the probability that my sample ...
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Period of linear congruential generator
This is a cross-post of the unanswered question (the given answer turned out to be incorrect) https://math.stackexchange.com/questions/245591/period-of-linear-congruential-generator .
How can you ...
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bound the hitting time in Markov chain
Given a finite-state Markov chain $M$ and assume there is a terminate state $f$ which is reached with prob. 1, I am interested in the distribution of hitting time $T$ of $f$, namely, $T$ is defined as ...
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How to solve a specific multivariate recurrence relation (or general ones)
How do you solve this recurrence (or multivariate recurrences in general)? Note that $p\in[0,1]$ and $n\in\mathbb{N}$ are given constants, where $np\leq 1$.
$$f:(\mathbb{N}\cup\{0\})\times(\mathbb{N}\...
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transform a polynomial into another one upto a constant
I have a polynomial $p(x)=a_Nx^N+a_{N-1}x^{N-1}+\dots+a_0$. I want to convert this into another polynomial of same order, say $b_Ny^N+b_{N-1}y^{N-1}+\dots+b_0$. Is it possible to find a transformation ...
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distinguishing random orthogonal matrix from Gaussian random matrix
Jiang's paper (http://projecteuclid.org/euclid.aop/1158673325) shows the following: Suppose that $G$ is an $n\times n$ random matrix with entries i.i.d. $N(0,1/n)$, and $Z$ is a random $n\times n$ ...
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How to work with infinite random graph(s) ?
Hi,
In the case where we are dealing with an infinite random graph (RG with infinite nodes).
How do we model/work with notions like degrees, degree distribution ? How are they defined ?
Thanks!
- 14k
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Practical way to check for geometric convergence
Target distribution is multimodal, 24 dimensions, continuous state space. For MCMC integration (MH sampler) I use a manually tuned proposal distribution.
When I measure the convergence rate ...
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1
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Backing into a distribution function from an infinite moment sequence
Let's say you are given that $E(X^n)$ = $\frac{n!}{((n+3!)/3!)}$ for a random variable $X$. So the first 4 moments are $\frac{1}{4}, \frac{1}{10}, \frac{1}{20}, \frac{1}{35}$, and so on. Is there ...
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Integration over the orthogonal group
Let $O(N)$ be the orthogonal group, and $a,b,c\in\mathbb N$. The question is:
$$\int_{O(N)}U_{11}^aU_{22}^bU_{33}^cdU=?$$
This is quite a tricky question:
(1) The first thought would go to ...
- 188k
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SDE-removal of the diffusion coefficients
from math.stackexchange
I'm currently looking at stochastic differential equations with irregular coefficients such as $W^{1,p}_{loc}$. If I have
\begin{align}
dX_t=b(X_t)dt+\sigma dW_t,
\end{align}
...
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Do distinct idempotent measures on finite binary systems have distinct supports?
Suppose that $(S,*)$ is a finite set equipped with a binary operation. Extend the binary operation to the vector space $V$ with basis $S$.
The set of probability measures on $S$, viewed as a compact ...
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Additive energy of random sets
Given two random sets $A,B$ in a finite field (say $x\in A$ independently and with probability $1/2$), what is known about the additive energy $E(A,B)=|\{(a,a',b,b')\in A\times A\times B\times B: a+b=...
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Equivalence between choosing a subspace and choosing its orthogonal
Hi,
We consider subspaces of $\mathbb{R}^N$.
Suppose that we have a property called $\mbox{Prop}$ that apply to subspaces of $\mathbb{R}^N$. That is to say a function from the set of subspaces of $\...
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Conditional probability with permutations
Hello,
This problem looks very simple and I conjecture it's true but I have a hard time proving it. It'd be very useful for my work (I'm doing a PhD) and I'll be glad to cite you in a future article ...
CommunityBot
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An optimization problem, non complete bipartite graph and hungarian algorithm
I have two tables at my disposal, one work dataset and one reference dataset. Each dataset has got two columns, lets say these are fields A and B. I would like the rows in reference dataset with the ...
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Extremal point and probability
Let $(X,\mathcal{F},\mathbf{P})$ be a probability space and $f \colon X \mapsto \mathbf{R}^n$ an integrable function. We assume that $f$ takes its values in a closed convex set $C$ of $\mathbf{R}^n$ ...
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Gaussian Valued Random Variables in Geometry of Banach Spaces
Why are Gaussian valued random variables so important in the Geometry of Banach spaces? I am reading the monograph by Pisier - "Probabilistic Methods in the Geometry of Banach Spaces" and in the very ...
- 17k
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520
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inequality for coupling of measures
Let $X = \prod _{s \in S} \Omega_s$, with $\Omega_s$ finite and all the same, $S$ countable. Let $\mu_1$ and $\mu_2$ be two probability measures on the product space (not necessarily the product ...
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coupling of projections and projection of the coupling
Let $C$ be a coupling between two measures, $C= \mu^1 \mbox{ } t \mbox{ } \mu^2$ ($t$ is the symbol of binary operator of the coupling (I can't find a more proper symbol here)). The measures are both ...
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The expected minimum Hamming distance within a set of randomly selected binary strings
If I randomly sample with replacement $P$ times from a set of all possible binary strings of length $L$, what is a good lowerbound on the expected minimum Hamming distance between any two of my $P$ ...
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Proving an asymptotic property regard the fraction of '1' and '0' in binary sequences [closed]
Hello,
Consider the set of sequences of zeroes and ones of length $N$ with $k$ ones (or, Np ones where $p=k/N$). We draw randomly and uniformly a sequence from this set.
I want to show that with ...
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Does martingale convergence hold for arbitrary time?
Let $\{\mathcal B_i:i\in I\}$ be a family of $\sigma$-algebras (over the same set $\Omega$) which are totally ordered by inclusion, in the sense that for any $i,j\in I$ either $\mathcal B_i\subset\...
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Trasportation metric (AKA Earth-Mover's, Wasserstein, etc.) as "natural" / "induced"?
Context: Given a discrete finite metric space $X$ (in my case X={0,1}$^n$ with the Hamming/L$_1$ distance), I need to define the natural or canonical metric on the set of all probability distributions ...
CommunityBot
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Limit of the stochastic process at time 0
This is not a homework question so please be kind not to remove it right away. I am working on some research but have to justify the following argument: Assume $S_t$ is a continuous stochastic process,...
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First order approximation of the current in ASEP
I am searching for an elementary proof of the first order approximation of the current of particles in ASEP (asymmetric simple exclusion process) and TASEP (totally asymetric). To avoid technical ...
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random matrix products reference
For a long time the standard (though not the easiest to find) reference on random matrix products was Bougerol and Lacrois:
Bougerol, Philippe, and Jean Lacroix. Products of random matrices with ...
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Properties of the Euler Discretization of a diffusion
Let $X$ be a continuous 1-d diffusion:
$$
dX_t = a(X_t)dt + b(X_t)dW_t, X_0 = x.
$$
W is a standard Brownian Motion and $a(\cdot)$ and $b(\cdot)$ can have nice regularity properties.
Let $Z^n_1,Z^...
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Derivative of the most probable value (of a gaussian variable) VS most probable value of the derivative
Let $x$ be a random variable with gaussian probability distribution $P(x)$. We assume that $x$ depends parametrically on a parameter $t$ so that :
$P(x(t))=\frac{1}{\sqrt{2\pi\sigma^2(t)}}\exp(-\frac{(...
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Expected Hitting Time for Simple Random Walk from origin to point (x,y) in 2D-Integer-Grid
Consider a simple random walk on the lattice $\mathbb Z^2$ starting at the origin $(0,0)$ where in each step, one of the four adjacent vertices in chosen uniformly at random, i.e. with probability $1/...
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Computing a density function for the integral of a stochastic process, given its transition function
$P$ is a one-dimensional Markov stochastic process that runs on time interval $[0, t_f]$. I know its transition function: $P(0) = x_0$ and for any $0 \le t_a < t_b \le t_f$, the function $f(x_b | ...
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Algorithm for numerically approximating the Prokhorov metric?
Question: What is known about algorithms for numerically computing/approximating the Prokhorov distance between two measures?
Recall that the Prokhorov distance metrizes the topology of weak(-*) ...
CommunityBot
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Extending Wald's equation to two classes of i.d. random variables?
I try to adopt Wald's equation to a slightly more complex problem. In fact, after a full day, I found some solution now, but it has a confusing argument in the middle. Perhaps somebody can help me at ...
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Inequality regarding $\ell_p$ norms, $p<1$
Let $(x_{i,j})$ be an infinite double sequence of nonnegative real numbers, and $ 0< p<1$.
I would like to know whether one can bound from above the sum
\begin{equation}
\sum_{i,j} x_{i,j}^p
\...
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Median-of-k elements
Hello,
Assume I am given a sequence of $n$ elements (by sequence I mean an ordered set). I want to randomly pick $k$ elements out of these $n$ elements, where $k$ is an odd number $\leq n$.
Then out ...
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Find a minimum entropy code for a simple gibbs random field.
Just to make precise what I am talking about, I will include the definition of a minimum entropy code. I will then define the precise markov random field I am asking about.
In the rest of this ...
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Sampling from a recursively defined distribution
I'd like to know if there are techniques for sampling from a recursively defined probability distribution, assuming that solving the recursion for a formula for the distribution is too difficult.
As ...
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Probabilty of two permutations having common elements?
What is the probability of two permutations on set X of size m (i.e. |X|=m) having at least n points of intersection? By this I mean that if two permutations, which I'll call g(x) and h(x), map a ...
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Linear Maps between $L^1$-spaces of singular measures
I posted the following question also here, but thought that I can get more answers in MO.
Let $(\Omega,\Sigma)$ be a measurable space and $\nu_1$, $\nu_2$ two probability measures on it. For $i=1,2$, ...
CommunityBot
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Lotteries, Turan's problem, and minimization of risk
Suppose I am a high-volume broker aiming to make some money on a state lottery. In this lottery, six balls are drawn from a population of (let's say) 50, without replacement. A ticket is a choice of ...
- 2,704
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What is the distribution of the distance between a specific word in a Text which is generated by a markov process?
What is the distribution of the distance between a specific word in a Text which is generated by a markov process?
For example for a text which is generated by a multinomial distribution over words, ...
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transition probability convergence for Harris chains - Durrett.
Dear mathoverflow.
This is a question to a proof in a graduate text. I have asked two professors at my university without help, so I hope it suffices in difficulty for this forum otherwise I ...
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Random parking problem on a probability distribution
Rényi's Parking Constants comes up when one puts down unit length cars on a interval, such that the probability of covering any two interval is the same.
Are there any published results when the ...
CommunityBot
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4
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A test for randomness of direction of vector data
I want to test the hypothesis that a group of vectors in 3D space, say given by a long list of xyz coordinates from some experiment, have no preferred direction. Is it sufficient to pick some ...
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Permutations & Balanced Distribution
I would like to implement a form of consistent hashing using a set of permutations.
The rules are as follows:
I have Y=~32 buckets and X items. Buckets may be "alive" or "dead". Items are to be ...
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