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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Concentration bounds for sums of random variables of permutations
I'm trying to find theorems regarding random variables derived from sampling permutations, specifically concentration bounds.
As an example, let $X_i$ be the $\{0,1\}$-random variable that represents ...
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1
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Compute the expected value of the product between a Lebesgue–Stieltjes type integral and an Ito integral
Hi, I have the following expected value to compute
$E[ \int_{o}^{T} f(t) dt \int_{o}^{T} H(s) dW(s)]$,
where $f(t)$ and $H(s)$ are two stochastic processes adapted to the filtration generated by the ...
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Mean value theorems for the Haar integral?
Let $G$ be a compact topological group (feel free to add hypotheses if necessary). Is there any mean value theorem for its (normalized to 1) Haar integral?
In general, are there mean value theorems ...
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Probability density that minimizes the sample range
Let $\mathcal{F}$ denote the set of all "concave probability distributions" on the unit interval, that is, all functions $f:[0,1]\to \mathbb{R}$ such that $f$ is concave, $f(x)\geq 0$ for all $x\in [0,...
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Convergence of Dirichlet Forms
If a sequence of Dirichlet forms convergence to 0, then what about the diffusion processes associated with these Dirichlet forms? Do the finite dimensional distributions of them converges weakly? and ...
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What is the most extreme set 4 or 5 nontransitive n-sided dice?
A set of nontransitive dice is a set of dice whose face numbers are such that the relation "is more likely to roll a higher number than" is not transitive. (See wikipedia)
For some sets, the ...
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Completion time of a process on a tree
Given is a constant degree rooted tree of depth $D$. It is also known that the total number of nodes in the tree is at most $D^2$. There is a probabilistic process with discrete time steps on the ...
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Does Multiplicative Version of Azuma's Inequality Hold?
It is known that there are multiplicative version concentration inequalities for
sums of independent random variables. For example, the following
multiplicative version Chernoff bound.
Chernoff bound:...
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Expectation of random matrix inverse
Given a $K\times M$ matrix $X$, where $M\gg K$, comprising independent complex Gaussian random variables, each one with mean
$$E[X_{k,m}]=B_{k,m}$$
and variance
$$Var[X_{k,m}]=\Sigma_{k,m}$$
define ...
3
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game theory - coin flipping question
Lets say 2 players A and B try to have the most money at the end after playing a casino game in which they have a $49\%$ chance to double a wager.
Here are the rules to the bet between A and B:
Both ...
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Probability of summing products of irreducible polynomials in a finite field to zero
Let $f(x), g(x), h(x)$ be randomly chosen irreducible polynomials over the finite field $GF(2^n)$.
What would the probability be for $\sum_{(i,j,k:i,j,k\in\mathbb{N},i+j+k=C)} f^i(x)g^j(x)h^k(x)=0$, ...
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How to generate a random (Weyl) curvature operator ?
Given a dimension $n$, the space of curvature operators is the space $S^2_B(\Lambda^2\mathbb{R}^n)$ of symmetric endomorphisms $R$ of $\Lambda^2\mathbb{R}^n$ which satisfy the first Bianchi identity :
...
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Probability of all combinations of k numbers among n being coprime
A simple argument shows that if we choose $n$ positive integers at random, the probability of their greatest common divisor being 1 is $1/ \zeta (n)$ (in the sense that if we choose the numbers among $...
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Randomly switching street lights, in a square city
This is a combinatorics-probability question, best stated however in "recreational" terms. Imagine a $N\times N$ city, meaning that we have $N$ horizontal streets, and $N$ vertical streets. At each ...
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Convexity of a Certain Set of Covariance Matrices
Hello,
My question is about a certain set of matrices being convex or not. I'll start with some preliminaries in order to define myself properly. Let $X_1,U,X_2$ be three zero-mean Gaussian random ...
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"Soft" Voronoi cells or statistical criterias
It is probably some basic statistics question, but...
Informally 1: How to choose "criteria", such that it will guarantee that error decision probability is less than "epsilon", and maximize ...
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Coupling of vectors
Let $X = (X_1,X_2)$ and $\hat X = (\hat X_1,\hat X_2)$ be two random variables where $X_i,\hat X_i$ are taking values over the Polish space $E_i$ endowed with their Borel $\sigma$-algebras, where $i=1,...
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What is the expected value for this
If there are $8$ random points in the plane whose horizontal coordinate
and vertical coordinate are uniformly distributed on the open interval
$\left(0,1\right)$, what is the expected largest size of ...
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2
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710
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Runs in coin flips
Let $P(j,k,n)$ be the probability of getting $j$ uniform runs of length $k$ from $n$ fair coin flips. What's the best way to compute $P$? I have no idea how difficult it might be; if it's a very ...
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Are numbers $h_{r,s} = \sum_{k} P(r;s;k) \frac{1}{n^{2k}} \bigg(1-\frac{1}{n}\bigg)^{n-2k}$ irrational?
I asked this question on MSE and Mike Spivey gave an insightful answer. I decided to put it here nevertheless in case someone else gets interested. If this violates rules on MO, please let me know, I'...
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Interesting thesis topic on statistical inference that is sufficiently mathematical
Hello , I am a student who's gonna start honours in mathematics . Currently , I am at the stage of finding a suitable honours thesis topic . I've chosen my supervisor , who's research interest is on ...
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Convergence rate for product of stochastic matrices
Hi,
I have a system of the form $$x(t+1) = A(t + 1) x(t),$$ for $t \geq 1$, and some fixed initial condition $x(1)$. Here $A (t)$ is a time-varying $m \times m$ matrix that is stochastic at all ...
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What is the probability that a random subset of a finite group is generic?
Definition 1: Given a group $G$, a subset $X \subseteq G$, and a natural number $k$,
we say that $X$ is (left) $k$-generic in $G$ if there are $k$ many left translates of $X$ that cover $G$.
That is, ...
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Topological conditions of Kolmogorov Extension Theorem
KET is often used to construct stochastic processes in continuous time when the state space is $\Bbb R^d$. As far as I am familiar with its proof, it uses standard monotonic class-like arguments ...
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Tails of sums of Weibull random variables
Suppose that $X_1, X_2, \ldots, X_n$ are i.i.d random variables distributed according to Weibull distribution with shape $0 < \epsilon < 1$ (it means that $\mathbf{Pr}[X_i \geq t] = e^{-\Theta(t^...
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The probability for a symmetric matrix to be positive definite
Let me give a reasonable model for the question in the title. In ${\rm Sym}_n({\mathbb R})$, the positive definite matrices form a convex cone $S_n^+$. The probability I have in mind is the ratio $p_n=...
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Prove an inequality related to moments
I am reading a paper and stuck with an inequality used in that paper.
$\varepsilon^n=(\varepsilon_1^n, \varepsilon_2^n,\ldots,\varepsilon_n^n)^T$ is a vector of i.i.d. random variables with mean 0 ...
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Maximal principle for stochastic heat equation
Consider $\partial_{t}u=\partial_{xx}u$ with Neumann boundary condition
$u_{x}(0,t)=u_{x}(1,t)=0$ and initial condition $u(x,0)=f(x)\geqslant0$.
Then up to time $T$, the maximal value of $u$ should be ...
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An elementary probability question
Let $X$ be a $d$-dimensional random vector distributed according to probability measure $D$. At least the second moment of the coordinates of $X$ is finite.
Consider $n+1$ samples $X_0, \ldots, X_n ...
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Hidden Markov: representing joint probability for set of observations as a product of two subset probabilities
Good day to everyone!
My question concerns Hidden Markov Models and is pretty basic. In one of the books ("Introduction to Machine Learning" by Ethem Alpaydin, 2nd Edition, p.373), I get the ...
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Is there a general process for conditioning a stochastic process above a boundary?
$(X_t, Y_t)$ is a two-dimensional Markov stochastic process that runs on time interval $[0, t_f]$. Given its transition function $a(x, y | x', y')$, I would like to condition the process on $\inf_{s \...
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Approximating a hitting time for some state using the stationary distribution?
Provided a random walk on a bounded interval, with step probabilities, $p$ and $q$ and a stationary distribution $\pi$, how "bad" of an approximation is to assume that the hitting time for a position $...
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Is there a relationship between Entropy of a fininte distrete probability distribution and the squre sum of the values of probability mass function of that distribution?
Sorry for the long title. What I mean is that for two vectors (a_1,...,a_n) and (b_1,...,b_n) with the property $a_i,b_i \geq 0 $ and $ \sum a_i =\sum b_i =1$.
If $ -\sum a_ilog(a_i) > -\sum ...
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Which limit to take as a key applied math decision
The Borel-Kolmogorov paradox refers to situations where non-uniqueness in the notion of conditioning on a set of measure zero leads to apparent contradictions. As a formal matter, one requires ...
3
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1
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small hyperworlds ?
The theory of random graphs, after the pioneering classic work of Erdős & Rényi, has come to prominence with many further refinements, most notably the small world theory (Barabási, Watts, etc).
...
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What is a random number? (poll experiment) [closed]
Imagine the following experiment: you wait say at a subway exit, and ask everyone passing "please tell me a number" (positive integer, of course). You do this day after day, until you reach say 1M ...
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Streamlined probability measure for tossing infinitely many coins
The standard probability measure over countably many independent coin tosses (i.e., the probability that you get a prescribed prefix of length $v$ is $2^{-v}$) is usually obtained via results in ...
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Applications of the Giry monad in probability and statistics
In another thread, I asked about the $M$ endofunctor on the category $\operatorname{Meas}$ of measurable spaces, which sends a space $X$ to its space of measures $M(X)$.
Will Sawin described the ...
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Fixed objects of the M endofunctor on category Meas
Consider the category $\operatorname{Meas}$ of measurable spaces: its objects are sets equipped with $\sigma$-algebras, and its morphisms are measurable functions between spaces.
As Gerald Edgar &...
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Is there a known asymptotic scaling for the probability of recurrence for a walk on $Z^d$?
I'm curious if there is a known asymptotic scaling for the return-to-origin (i.e. recurrence) probability for a random on $Z^d$ as a function of $d$?
Mathworld gives the recurrence probability:
...
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1
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Estimates for the mixing time of a Markov Chain with biased initiation
Imagine I have some Markov process consisting of a biased random walk on the integers, over some interval $[0, L]$, with $+1$ and $-1$ step probabilities of $p$ and $q$, respectively, s.t. $(p + q) = ...
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a problem on DTMC
For a Markov chain $\lbrace X_n, n\ge0\rbrace$ with transition probabilities $P_{i,j}$, consider the conditional probability that $X_n = m$ given that the chain started at time $0$ in state $i$ and ...
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What is the purpose of The Four Moment Theorem for Complex Matrices? [closed]
How is this theorem useful in the real world? In general, what is the point of studying limiting distributions of various statistics of random matrices?
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When is a space of measures a measurable space?
Let $X$ denote a measurable space, that is, a set equipped with a $\sigma$-algebra $\Sigma(X)$. Let $M(X)$ denote the space of real-valued measures over $X$. This is a vector space over the real ...
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The human body's random number generator
I remember learning in microbiology that the human body generates antibodies using a random process so that an enormous variety of antibodies can be produced with a simple genetic code.
Now that I'm ...
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Multinomial -- how many trials in order to see all the values with prob 1-\alpha
Let suppose that I have a box with $k$ different balls, each one with a different color.
At each time I have to extract a ball and observe the color. Then I put the ball back in the box.
How many ...
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Arithmetic properties of erf functions
I was messing around with Benford's law trying a proof to fill up time on a Saturday, and I ran into a problem. I have the equation $\frac{\mathrm{erf}(2x)-\mathrm{erf}(x)}{\mathrm{erf}(10x)-\mathrm{...
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3
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Support of an infinitely divisible measure.
Hello,
if $G$ is a compact Lie group. Let $\mu$ be an infinitely divisible measure on $G$, such that $e$, the neutral element of $G$, is in the support of $\mu$. Is that true that the support of $\...
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Do the Eigenvectors find by use PCA on a set of data point, a good replacement for Random Projection when I later on use L1Magic to reconstruct the sparse vector?
Concretely if I use the first k eigenvectors find by PCA with a point set A,to project another sparse vector b to k dimension subspace, then use L1-magic to recover b. Will this be better than a ...
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The Fourier Transform of taking Eigenvalues
The purpose of this question is to ask about the Fourier transform of the map which associate to an $n$ by $n$ matrix its $n$ eigenvalues, or some function of the $n$ eigenvalues. The main motivation ...
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