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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Fourier transform of distributions with non-standard test functions
This might be a quite simple question for function analysis standards, but it has some obstacles. I'll try to improve the readability a bit by not using the full tex code. A short motivation:
Given a ...
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Randomly contracting edges of a graph - expected number of vertices?
Let $G'$ be a graph obtained from $G$ after contracting each edge with probability $p$. Let $n = |V(G)|, e = |E(G)|$.
I would like to compute (or at least obtain a lower bound) for $E[|V(G')|]$ in ...
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Is there a corresponding Hahn decomposition theorem for the real-valued Radon measures?
Hello,
As we know that a signed measure $\mu$ on $R$ can be decomposed to the positive part $\mu_+$ and negative one $\mu_-$ by the Hahn decomposition theorem.
My question is whether each real-...
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Product of correlated random variables
Suppose $X_1,...,X_n$ is a sequence of stationary correlated random variables in $\{-1,+1\}$ such that :
$\mathbb{P}[X_i = +1] = p$,
$\mathbb{P}[X_i = -1] = 1-p$ with $p\in (0,1)$,
and with a ...
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Can singular measures be viewed as vanishing distributions? (Answer No!)
Hello,
Here is my original question: let $\mu$ be a singular measure with respect to the Lebesgue's measure on $R$. Is it true that $\int \psi \mu(d x)=0$ for any test function $\psi\in C_c^\infty(R)$...
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Puzzle in Martin Gardner book [closed]
What is the official name of this problem? Martin Gardner gives introduction in his book "Math circus". The problem belongs to 1D random walk. What can be read to gain deep insight into this problem? ...
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An infinite Gaussian mixture with mixing parameters being also Gaussian
A finite Gaussian mixture with $k$ components has a probability distribution function $p(y|\mu_1,...,\mu_k, \sigma_1, ..., \sigma_k, \pi_1, ..., \pi_k)=\sum_{j=1}^{k} \pi_j\mathcal{N}(\mu_j, \sigma_j^...
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Compact sets of the complex plane having the K-property ?
I would like to have a better understanding of a notion I've met in the beautiful book of Nikishin and Sorokin "Rational approximation and Orthogonality", since they do not provide examples.
As it is ...
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Did Joseph Doob prove that random sequences don't exist?
In the book "The Mathematical Experience" it says:
"An infinite [binary] sequence $x_1, x_2, \ldots$ is called random in the sense of von Mises if every infinite sequence $x_{n_1}, x_{n_2}, \ldots$...
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Weak*-continuity of regular conditional probabilities "in time"
Let $(\Omega, F, (F_t)_{t\geq 0}, \mathbb{P})$ assume that $(X_t)_{t\leq T} $ is some cadlag, real valued stochastic process, not too bad: say something like a Brownian Motion and some Poisson finite ...
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Number of real solutions of a random equation
Let $(J_{ij})$ be an $n \times n$ random matrix with i.i.d Gaussian centered coefficients with $\displaystyle \mathbb{E}[J_{ij}^2] = \frac{\sigma^2}{n}$.
Let the random variable $A_n(\sigma)$ defined ...
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BM and interpretation of stopping time sigma algebra
Suppose $H$ and $K$ are open subsets of $\mathbb{R}^d$ containing the origin with $H\subset K$, $B_t$ a standard Brownian motion starting at the origin, $\mathcal{F}_t$ its canonical filtration, and $\...
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name for $\underset{x}{\operatorname{argmin}} \displaystyle\sum\limits_{i=1}^n |x_i - x|$
Given a real-valued data set $ x_1, \dots, x_n $, what do you call the quantity
$$\underset{x}{\operatorname{argmin}} \displaystyle\sum\limits_{i=1}^n |x_i - x|$$
This seems like a pretty basic ...
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Reading Material on Couplings
Does anybody have suggestions on what to read to learn more about couplings pertaining to statistics?
I'm working on a research project on Poisson approximations and am looking to perform a coupling ...
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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 ...
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Who is the weak* sequential closure of the set of finitely supported measures on the integers?
Let $X$ be a topological space and $Y\subseteq X$, the sequential closure of $Y$ is the set of elements in $X$ that are limit of sequences belonging to $Y$.
Let $\mathcal M_{\text{fin}}(\mathbb Z)$ ...
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law of iterated logrithm
Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which
$\mathcal{F}_t$ is filtration satisfying general conditions. $W_t$ is
a standard Brownian motion. By law of iterated logarithm, one has
$...
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Marginals and Convex Sets
I am looking for a weak convergence of marginals result, in the following type of situation. References to 'related' situations are also very much appreciated.
I have a collection of affine ...
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Bounding mutual information given bounds on pointwise mutual information
Suppose I have two sets $X$ and $Y$ and a joint probability distribution over these sets $p(x,y)$. Let $p(x)$ and $p(y)$ denote the marginal distributions over $X$ and $Y$ respectively.
The mutual ...
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Standard way of determining if you have enough data to reliably compute success probability
Given $s$ successes in $n$ trials, where $p=\frac{s}{n}$, is there a standard way to determine if I have enough data to compute a meaningful statistic? For example, given $s=1, n=10, p=0.1$, the 95% ...
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Efficient computation of $E\left[\frac{1}{1+\sum_iX_i}\right]$ where $X_i$ is RV with Bernoulli distribution with different probabilities
Suppose we have the random variables $X_1, \ldots, X_n$ that have Bernoulli distributions with the (possibly different) probabilities $p_1, \ldots, p_n$. For example, $X_1$ = 1 with probability $p_1$ ...
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Heavy Tailed Network
In his paper Kronecker Graphs: An approach to modeling Networks Jure et Al, mention that an important property of networks are that they are heavy tailed.
I'm trying to get an insight on what this ...
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Estimating the Distribution of a Very Large Population of Known Size and Unknown Variance
I would like to estimate the distribution of a very large population of known size but unknown mean and variance. I cannot assume anything about the underlying distribution. The values of observations ...
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Probability and information. The burrel-bucket-glass problem.
Suppose we have a barrel with three different kinds of marbles: red, green and blue. The probability to find a red marble in the barrel is R0, analogously the probability for green is G0 and for blue ...
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The expectation of $\sqrt{B(n,p)}$
Let $n$ be a growing integer parameter, and suppose that $X_1,\dotsc,X_n$ are independent Bernoulli random variables with the probabilities of success $p_i:={\mathsf P}(X_i=1)$. If $X=X_1+\dotsb+X_n$ ...
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Kullback-Leibler divergence of scaled non-central Student's T distribution
What is the Kullback-Leibler divergence of two Student's T distributions that have been shifted and scaled? That is, $\textrm{D}_{\textrm{KL}}(k_aA + t_a; k_bB + t_b)$ where $A$ and $B$ are Student's ...
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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 ...
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Cover time for a biased random walk on an 'N'-dimensional integer lattice
Imagine that I have a random walk on an $N$-dimensional integer lattice, $Z^N$, of finite dimensions, $(d_1, ..., d_N)$, where boundaries are fully reflecting and the walker is initialized at some ...
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A question on random walks on semisimple groups
Let $G$ be a connected semisimple Lie group without compact factor, $\mu$ be a Borel probability measure on $G$ such that the group generated by $\mathrm{supp}(\mu)$ is Zariski dense in $G$. For ...
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# bridges in random connected graph
Suppose we have an Erdos random graph with $n$ vertices and $c n$ edges.
What can you say about the probability that the graph is connected?
(More importantly) If it is connected, what is the ...
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Probability formula derivation
How is the following formula derived which yields the probability that the sum of the squares of n random draws from the closed interval [-1,1] is less than one?
formula: (1/2^n)*pi^(n/2)/(n/2)!
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Is there a probabilistic interpretation of Dedekind zeta functions?
Reading the interesting paper Honest Bernoulli excursions by Smith and Diaconis motivated the question whether probabilistic interpretations for general Dedekind zeta functions are known.
In the ...
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How can the topological entropy and $L^2$ mixing rate be related?
For a product of otherwise identical systems evolving at different rates, the toplogical entropy and a quantity very closely related to (indeed, identifiable with a nondegenerate variant of) the $L^2$ ...
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exactly simulating a random walk from infinity
In diffusion-limited aggregation on the square lattice, one lets a particle do "random walk from infinity" until it hits the current aggregate, at which point the site occupied by the particle is ...
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How to obtain tail bounds for a linear combination of dependent and bounded random variables?
Hi everyone,
Note: This question is a general case and edited version of my previous question ``How to obtain tail bounds for a sum of dependent and bounded random variables?''.
I am looking for ...
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Covariance sign
Hi,
Is it true that $Cov[f(X),g(X)]>0$ where $X$ is a random variable of unbounded support and $f,g$ are two strictly increasing real functions? I think by Chebyschev integral inequality I must ...
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"Reverse" stochastic dominance
Let $\mu$ and $\mu'$ be probability measures on $\lbrace0,1\rbrace^\Lambda,\:\: \Lambda:= {\lbrace 0,1,\ldots,n\rbrace}$. Assume that
$\mu(X_i=1|X = \zeta \text{ on } \Lambda \setminus \lbrace i\...
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Mean of an experiment
Suppose we have a bag of n different balls, and each time m (m<n) balls are taken out for checking from the bag and put back. ...
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Conditional distribution of the modulus of the output of AWGN channel given the modulus of the input
Hi everyone,
I will be too happy if anybody help me find a solution for the following problem.
In fact, I have a big problem that I could not solve it for weeks.
Assume that we have we have two ...
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Sufficient conditions for independence based on moments
Let $P$ be the joint distribution of two random variables $X$ and $Y$, that both have support on $(0,1)$ (I am also interested in the case where $X$ takes values on $k$-dimensional simplex, but I ...
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Minimum Mean Square Error (MMSE) and Mutual Information (I)
Consider this setting:
$Y=X+N$
where $N$ is a Gaussian standard random variable and $X$ is another arbitrarily distributed r.v. You can think of this $X$ as a message being transmitted over an AWGN ...
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A M/M/$\infty$ queue of depositors with compound interest
Hello, I'm trying to model a bank's liabilities using a queue. Suppose a bank begins with a cash reserve of $M$. Depositors are a M/M/$\infty$ queue; they arrive with rate $\lambda$ and deposit 1 ...
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Correcting bias in samples selected by a prediction
Here is the scenario:
I'm trying to find as many golden tickets as I can, so that I can sell them to kids that want to go on a tour of Wonka's chocolate factory.
Fortunately, I have a machine that ...
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Probability inequalities
Hi everyone,
I am looking for some probability inequalities for sums of unbounded random variables. I would really appreciate it if anyone can provide me some thoughts.
My problem is to find an ...
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Average Hamming distance between strings after some number of random substitutions in a population of initially identical elements
Let's say I have a set $S$, $(s_1, ..., s_i, ..., s_P) \in S$, of $P$ identical strings over a $k$-letter alphabet, each of length $|s_i| = L$. With uniform random probability across all strings in $...
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Are affine continuous functions on Bauer sub-simplices of the probability measures given by integration over continuous functions?
Let $X$ be a compact (non-metrizable) Hausdorff space and $\mathcal{P}(X)$ the set of Radon probability measures with
weak-$*$ topology (weak topology induced by the continuous functions).
Consider a ...
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Green function of simple random walk
Let $G$ be the Green function of the simple random walk on $\mathbb{Z}^d,\:d\geq 3$; i.e.
$$G(x) = E \sum_{i=0}^{+\infty} 1_{X_i=x},$$
where $X$ is the simple random walk starting from $0$. The ...
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Intrinsically measurable subsets of amenable semigroups.
This question is related to the one in https://mathoverflow.net/questions/65322/the-structure-of-certain-maximal-sets-of-means-into-amenable-semigroups. I open a different topic because they can be ...
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To what extent can the following zero-one laws be relaxed?
I am interested in what circumstances various zero-one laws in probability theory can be relaxed. In particular, independence is a very important factor in such laws.
1) Borel-Cantelli Lemma: Let $...
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What results would follow from or imply "randomness" of the primes?
This question on random versions of deterministic problems reminded me that many conditional results in number theory hold if the primes are in some sense random, and it is common knowledge that the ...
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