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.
9,022 questions
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Inequality in Gaussian space -- possibly provable by rearrangement?
The following problem arose for my collaborators and me when studying the computational complexity of the Maximum-Cut problem.
Let $f : \mathbb{R} \to \mathbb{R}$ be an odd function. Let $\rho \in [...
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How does the Dirichlet process work?
Hi, i'm looking to get into nonparametric bayesian techniques but I'm having problem understanding what's going on in the definition of the Dirichlet process or how it works. So what does P ~ DP(&...
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How is it that you can guess if one of a pair of random numbers is larger with probability > 1/2?
My apologies if this is too elementary, but it's been years since I heard of this paradox and I've never heard a satisfactory explanation. I've already tried it on my fair share of math Ph.D.'s, and ...
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Probability vertices are adjacent in a polygon
With regard to my original question:
A subset of k vertices is chosen from the vertices of a regular N-gon. What is the probability that two vertices are adjacent?
I suppose that the responses ...
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Probability Question
You have $N$ boxes and $M$ balls. The $M$ balls are randomly distributed into the $N$ boxes. What is the expected number of empty boxes?
I came up with this formula:
$\sum_{i=0}^{N}i\binom{N}{i}\...
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Result of repeated applications of the binomial distribution?
What is the result of multiplying several (or perhaps an infinite number) of binomial distributions together?
To clarify, an example.
Suppose that a bunch of people are playing a game with k (to ...
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A variant of the hypergeometric distribution - in the literature?
I have been working on a problem in combinatorics that makes use of the following discrete distribution.
Let $a_{1}, a_{2},..., a_{N}$ be any binary sequence of of length $N$ with $n$ ones and $m$ ...
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Monte Carlo simulations
I was wondering what were the models of statistical physics that are still considered difficult/slow to simulate (exactly, or approximately) with the current technology of Monte Carlo approaches. I ...
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A Local CLT with large variance
For n an even integer, $0 \leq i \leq$ ${n} \choose{j}$, $1 \leq j \leq n$ let $X_{i,j}$ be a
random variable taking values $\frac{n}{2}-j,0,j - \frac{n}{2}$ with equal probability. Let $S_{n}$ be ...
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Coupling of Wiener processes
Is it possible to find a coupling of two Wiener processes $W^0, W^x$ (i.e. two Wiener processes defined on a common probability space). One starting from $0$ and the other from $x$ such that
$W_t^0 -...
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A formula for moments of the limit distribution of singular values in the proof of the circular law
One of the steps in the proof of the circular law in random matrix theory is obtaining the limiting spectral distribution for the matrix
$(\frac{1}{\sqrt{n}} X_n - zI)(\frac{1}{\sqrt{n}} X_n - zI)^\...
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Martingales and Betting Strategies
Does anyone know of a good introduction to the theory of martingales and betting strategies from the point of view of statistics and/or probability theory? I'm looking for something basic, with lots ...
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Models with SLE scaling limit
What discrete processes/models have been proven to have scaling limits to $\text{SLE}(\kappa)$, for various $\kappa$?
I know about loop-erased random walk and uniform spanning trees.
What about ...
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Conditional expectation of convolution product equals..
Let $X, Y$ be two $L^1$ random variables on the probablity space $(\Omega, \mathcal{F}, P)$. Let $\mathcal{G} \subset \mathcal{F}$ be a sub-$\sigma$-algebra.
Consider the conditional expectation ...
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Are two probability distributions uniquely constrained by the sum of their p-norms?
Let A, B and C be finitely supported probability distributions with at most d nonzero probabilities each. Now consider the following simultaneous equations using p-norms, for each value of p≥1, ...
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measurable sets not depending on even coordinates
Let $A\subset\{0,1\}^\omega$ be a measurable set (w.r.t. the usual borel sigma algebra) which does not depend on any even coordinate (that is, if $x\in A$ and $x$ and $y$ agree except on a finite ...
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most general way to generate pairwise independent random variables?
Is there a sort of structure theorem for pairwise independent random variables or a very general way to create them?
I'm wondering because I find it difficult to come up with a lot of examples of ...
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A probabilistic inequality [closed]
Suppose $x_1,x_2,...,x_6$ are non-negative Independent and identically-distributed random variables, is it true that $P(x_1+x_2+x_3+x_4+x_5+x_6 \lt 3\delta) \lt 2P(x_1 \lt \delta)$ for any $\delta \...
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Chances to win an election
Let's say that tomorrow national president election is held. A poll asks 1100 persons which of the two candidates, A or B, will he or she will vote. 750 say will vote A, and 250 say will vote B. What ...
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What's the standard name for sets of a given size with maximal probability (or a given probability and minimal size)?
The definition I'm going to give isn't quite the concept I really want, but it's a good approximation. I don't want to make the definition too technical and specific because if there's a standard name ...
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Parity, Balls and Boxes
Start with a distribution $\mu$ on [n], and drop m balls into these n+1 slots independently and according to the distribution &mu. That is, we have iid random variables x 1 through x m ...
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Difference Equations & Possible Limits
The answer to this may well be in some elementary textbook - a reference might be more useful than a short answer here.
If we look at the behaviour of a point in R n under matrix multiplication, we ...
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A problem of an infinite number of balls and an urn
I think that the following problem originated in a probability textbook :
You have a countably infinite supply of numbered balls at your disposal. They are all labeled with the natural numbers {1,2,3,...
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Intuitive explanation to Probability question [closed]
I have \$3. I flip a coin. If I get heads, I get \$1. If I get tails, I lose \$1. The game stops when I have \$0 or \$7. What is the probability I get \$7?
I solved this by creating a system of ...
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Which iid variables give a normal variable, when multiplied?
Hello, I hope you'll find my riddle interesting.
Z = XY
Z ~ N(0, 1)
X, Y are iid random variables (independent, identically distributed). We assume X and Y are symmetric.
What is the distribution of ...
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2
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Chances of streaks in small bit-streams
Let's say a series of 10 bits is output randomly. Now lets do that 256 times. I'd like to find out what the expected number of streaks of 1s or 0s are for each of the possible sizes 1-10.
For example,...
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Computing correlation between time series with missing data.
Suppose you have two simple Ar[1] series of the form $y_n=y_{n-1}+e_n$ and $x_n=x_{n-1}+m_n$, where $e_n$ and $m_n$ are normal white noise processes with no auto-correlation and $Corr(e_n,m_n)=p$. ...
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Formula for the nth convolution of a Laplace random variable
Let $x_1, x_2, ...$ be i.i.d. draws from a Laplace distribution with scale parameter $b$. Is there a relatively nice closed form for $x_1+x_2+...x_n$? I've seen a derivation floating around for when $...
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How long for a simple random walk to exceed $\sqrt{T}$?
Let $R_n$ be a simple random walk with $R_0 = 0$, and let $T$ be the smallest index such that $k\sqrt{T} < |R_T|$ for some positive $k$.
What is an expression for the probability distribution of $...
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order statistics for components of a random unit vector
Suppose you sample uniformly from the unit vectors in R^n. What are the distributions of the order statistics of the magnitudes of the components of the sampled vectors? That is, for 1 <= i <= n ...
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Expected value of a gamma-distributed random variable to the n-th power?
Is there a closed form for $E(Y^n)$, where $Y$ is a random variable with a gamma distribution with parameters $\alpha$, $\beta$?
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Harmonic mean of random variables
The arithmetic mean of normal random variables is normal. The geometric mean of log-normal random variables is log-normal. But is there a common distribution family closed under taking harmonic means?
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"The" random tree
One time I heard a talk about "the" random tree. This tree has one vertex for each natural number, and the edges are constructed probabilistically. Connect vertex $2$ to vertex $1$. Connect vertex $3$ ...
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Estimating probability of set membership
I have a number of discrete finite sets, $A_0$ through $A_n$. I do not actually know their contents, but I know the size of each set and the size of the intersection between $A_0$ and each of the ...
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What is convolution intuitively?
If random variable $X$ has a probability distribution of $f(x)$ and random variable $Y$ has a probability distribution $g(x)$ then $(f*g)(x)$, the convolution of $f$ and $g$, is the probability ...
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Deconvolution of gamma distributions
If the sum of two independent random variables is gamma distributed does this imply that the individual random variables are also gamma distributed. I suspect that the answer is no, but I do not know ...
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CLT for stationary sequences with infinite variance
There is a well-known central limit theorem for as a stationary sequences.
If $( X_n )_n$ is a stationary sequence and $E X_n=0$ then under suitable mixing conditions the sequence $S_n := n^{-1/2}\...
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Examples of random variables
I'm looking for a list of examples of random variables to use in teaching a measure-theoretic probability course. For example, the Rademacher functions are an explicit construction of independent ...
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When does a pointwise CLT hold?
Let $X$ be a random variable with mean $0$ and variance $1$, and let $X_1, X_2, X_3, \dots$ be iid copies of $X$. Under what conditions can we say that the density of $\frac{X_1+\dots+X_n}{\sqrt{n}}$ ...
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Laplace transform and fractional moments
Is there any "easy" way to calculate fractional moments from Laplace transform.
To be more specyfic let us consider the following example. Let $X$ be a positive random variable and
$L(\theta)...
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Evaluate a fair game [closed]
I'm not a mathematician, so my question may be not so clear, sorry.
Let's say we toss up an ideal coin and win 1 dollar on heads and lose 1 dollar on tails. So, expected value is M = 1×0.5 &...
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Is the min function ever an unbiased estimator for the mean?
Given $n$ i.i.d. variables $X_1$ to $X_n$ with an unknown probability distribution, the sample average is an unbiased estimator for the mean of the distribution. Is there some non-trivial probability ...
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When do binomial distributions occur?
A binomial distribution is the distribution of the number of successes of n independent, identical Bernoulli trials. What happens when the trials are dependent and the Bernoulli trials are not ...
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What is hidden in Hidden Markov Models? [closed]
Why the word "hidden" present in hidden markov model? What exactly is hidden.
Whatever is hidden in HMM isn't it hidden in normal Markov Models?
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How Does Random Noise Typically Look?
How does random noise in the digital world typically look?
Suppose you have a memory of n bits, and suppose that a "random noise" hits the memory in such a way that the probability of each bit being ...
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Singularity of sparse random matrices
The following topic came up in conversation with my office-mate Lionel: Let $p$ be a fixed prime, $c$ a fixed positive real parameter and $n$ a large number. Consider a random $(0,1)$ matrix with ...
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How many dimensions is it safe to get drunk in?
In Michael Lugo's blog post Variations on the drunken-bird theorem, and real-world sightings he wonders (without coming to a conclusion) what the maximum 'safe' number of dimensions to get drunk in ...
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Suprema of stochastic processes
Let X be a continuous stochastic process. I know that (t>s)
P(|X(t) - X(s)|>δ) < |t-s|/δ
Is it possible to say anything (e.g. estimate the decay of the tail) about
Y=sup_{s \in [0,1]} |...
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question on sigma-fields
Let X,Y to be mappings from the sample space Ω to R and suppose Y is measurable with respect to σ(X), the smallest σ-field that makes X measurable.
Does it follow that there exists ...
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Decoupling lemma for the Lambda(p) problem
I'm attempting to work through Bourgain's paper "Bounded orthogonal systems and the $\Lambda(p)$-set problem". There is a step in the proof of the decoupling lemma that I am stuck on, and thought ...
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