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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Monte Carlo method and possible applications to computer poker?
I want to do something about ”games of incomplete information“，like "Computer poker program".I know,Albert university(in canada) have do a lot of things to that field,they write a program called: "...
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Extreme value theory
I have been asked to provide an "approximation at infinity" of an expression that at the end simplifies to $-\frac{b e^{-a t}-a e^{-b t}}{a-b}$, in a course about extreme value theory.
In the course, ...
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Nice way to parametrize a bunch of non-independent discrete random variables
I'm looking for a "nice" way to parametrize the joint distribution of multiple, possibly correlated discrete random variables on {0,1}. Even for N=2, there doesn't seem to be an obvious way to do it. ...
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The probability for a sequence to have small partial sums
The question
Let $a_1,a_2,\dots,a_n$ be a sequence whose entries are +1 or -1. Let t be a parameter. My question is to give an estimate for the number of such sequences so that
$|a_1+a_2+\dots ...
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Examples of Slowly Mixing Chains in Statistics
This should probably be community wiki, but I don't know how to set that myself.
I'm looking for examples or Markov chains that are used in statistics or statistical physics, and which are known to ...
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What's the probability that k + n^2 is squarefree, for fixed k?
While playing around with this question (when is the sum of two squares squarefree?), from some experimental computations (and bolstered by the fact that the density of squarefree positive integers is ...
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How to show that an infinite sequence is normal if and only if every block of equal length appears with equal frequency?
An infinite sequence is normal if all strings of equal length occur with equal asymptotic frequency.
Formally, let $\Sigma$ be a finite alphabet of $b$ digits. Let $S$ be an infinite sequence and $\...
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Big Picture: What is the connection of Malliavin calculus with differential geometry?
I know that Paul Malliavin was heavily influenced by ideas from differential geometry while developing his calculus on Wiener space. But what are the concrete analogies between both areas of ...
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A geometric interpretation of independence?
Consider the set of random variables with zero mean and finite second moment. This is a vector space, and $\langle X, Y \rangle = E[XY]$ is a valid inner product on it. Uncorrelated random variables ...
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Convergence of Gaussian measures
Let $X$ be a separable Banach space with its Borel $\sigma$-algebra $\mathcal F$. Let $x_n \to x$ in $X$. Fix a Gaussian covariance operator $K$, and let $\mathbb P_n$ and $\mathbb P$ be Gaussian ...
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Finitarily Markovian Finite Factors of Bernoulli Schemes
By processes, I mean discrete, stationary stochastic processes, that is $(X,\mathcal{U},\mu,T)$ where $X$ is the set of doubly infinite sequences of some alphabet $A$, $\mathcal{U}$ is the $\sigma$-...
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Minimum Hamming Distance Distribution in a Random Subset of Binary Vectors+
Select $K$ random binary vectors $Y_i$ of length $m$ uniformly at random.
Let the following collection of random variables be defined: $X_{i,j}=w(Y_i \oplus Y_j)$ where $w(\cdot)$ denotes the Hamming ...
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Binomial distribution parity
Let $X \text{~} \text{Binomial}(n, p)$.
What is $\text{P}[X \mod 2 = 0]$? Is it of the form $1/2 + O(1/2^n)$?
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Generalized Cox Theorems, valuations on boolean sets, bayesian probabilities and posets
Bayesian probabilities are usually justified by the Cox theorems, that can be written this way:
Under some technical assumptions (continuity, etc, etc...), given a set $P$ of objects $A, B, C, \ldots$...
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Probability of n k-sided dice showing exactly m different faces
I found the following closed form solution for the abovementioned problem:
$${1\over k^n}\cdot{k!\over (k-m)!}\cdot{\{{n\over m}\}}$$ with ${\{{n\over m}\}}$ being the Stirling Number of the second ...
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Quantum analogue of Wiener process
The Wiener process (say, on $\mathbb{R}$) can be thought of as a scaling limit of a classical, discrete random walk. On the other hand, one can define and study quantum random walks, when the ...
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Probability of one binomial variable being greater than another.
I need to calculate (or bound) the probability that one binomial variable is greater than other. Specifically, if $x \leftarrow B(n,p)$ and $y \leftarrow B(n,q)$, what is the probability that $y \...
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Disintegrations are measurable measures - when are they continuous?
This is a sequel to another question I have asked.
The notion of disintegration is a refinement of conditional probability to spaces which have more structure than abstract probability spaces; ...
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When does a probability measure take all values in the unit interval?
Let $\mathbb{P}$ be a probability measure on some probability space $(\Omega,\mathcal{A})$. Are there conditions on the $\sigma$-algebra $\mathcal{A}$ such that for every real number $c\in [0,1]$ we ...
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Can we extract information about how fast a function decay from its Laplace transform?
My question is whether we can extract information about how fast an integrable function converges to zero by looking at the asymptotics of its Laplace transform.
More concrete case, let $f:\mathbb{R} ...
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How to fill a simplex with almost disjoint cuboids?
There is an algorithm that give us cuboids in $\mathbb{R}^3$, say $Q_1,Q_2,\ldots$, such that $\cup_{i=1}^{\infty} Q_i$ is the simplex with vertices $(0,0,0), (1,0,0) , (0,1,0), (0,0,1)$, and the $Q_i$...
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MicroArray, tesing if a sample is the same with high variance data.
I'll explain the problem but what I am looking for is a few suggested methods to approach this problem.
You don't need to know what a microarray but if you are interested look here link text
The info ...
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Notions of "independent" and "uncorrelated" for subsets of the natural numbers
In probability/statistics, there is a notion of two things being "independent", which would basically mean that any information we can get about one thing has no effect on our (probabilistic)...
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Distribution of the sum of the $m$ smallest values in a sample of size $n$
Let $\mathbf X = [X_1, X_2, \ldots, X_n]^\mbox{T}$ be a vector random variable drawn from a known distribution with CDF $F(x)$. The CDF for the minimum value in $\mathbf X$ is clearly $P[\min_{i=1\...
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Peakedness of multimodal distributions
In Probability theory, does there exist some measures of peaked-ness for multi-modal distributions. I guess kurtosis as such would not be a good measure of peaked-ness for multimodal distributions. ...
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Markov chain on groups
Let $G$ be a permutation group on the finite set $\Omega$. Consider the Markov chain where you start with an element $\alpha \in \Omega$ chosen from some arbitrary starting probability distribution. ...
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Hitting times for an N-dimensional random walk on a lattice with (strictly positive) random integer steps
Please consider a random walk on a finite N-dimensional lattice with vectors $(x_1, ..., x_N)$. We define the origin to be $(0, ..., 0)$ and the target to be at the point in the lattice furthest away ...
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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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Dense inclusions of Banach spaces and their duals
This seems like a really simple question, but I'm struggling with it. Let $X$ be a separable Banach space, $H$ be a separable Hilbert space, and suppose $i : H \hookrightarrow X$ is a dense, ...
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Random Alternating Permutations
An alternating permutation of {1, ..., n} is one were π(1) > π(2) < π(3) > π(4) < ... For example: (24153) is an alternating permutation of length 5.
If $E_n$ is the number of alternating ...
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probability mass function fitting [closed]
I have a probability mass function of some experimental data who's log looks like the following: (please ignore the fact that it is not normalized)
![alt text][1]
[image shack image removed]
(...
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Are there nonequivalent randomnesses?
There are nonequivalent geometries, nonequivalent groups finite and infinite, nonequivalent logics ( fregean and nofregean http://www.formalontology.it/suszkor.htm), even nonequivalent logicians;-)
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Exploding Levy processes
Hi,
probably this is a fairly newbie question, but is it possible that the a generic Levy process explodes (i.e. tends to infinity for finite time t with positive probability)? If yes, could you ...
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Has the Lie group preserving a probability distribution been used in Bayesian statistics?
For a (possibly signed) nondegenerate probability measure $\pi$ on $\{1,\dots,n\}$ define
$$\langle \pi \rangle := \{R \in \operatorname{STO}(n): \pi R = \pi \}.$$
Here $\operatorname{STO}(n)$ denotes ...
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Density of numbers having large prime divisors (formalizing heuristic probability argument)
I want to prove that the set of natural numbers n having a prime divisor greater than $\sqrt{n}$ is positive.
I have a heuristic argument that this density should be $\log 2$, which is approximately ...
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Joint Law with 2 marginals and marginal of the spread
I have a question for you and thank you in advance for your answers and ideas.
Let us suppose that we have the marginal distributions of two r.v X and Y, and also the law of X-Y (or any linear ...
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statistical approach to multinomial distribution
Suppose a dice with $q$ faces is rolled $N$ times, where $N$ is very big.
We define a multinomial variable $X=(X_1,\ldots,X_q)$ which counts how many times any face is occurred ($X_i$ is the number ...
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Asymptotics of symmetry types of tensors
Introduction
Let's fix $m\in \mathbb N$. For each n, the unitary group $\mathbf U(m)$ is represented in the space of tensors of rank $n$ over $\mathbb C^m$
$$V_{n,m}=\bigotimes_{k=1}^n \mathbb C^m$$ ...
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On operator ranges in Hilbert & Banach spaces
Lemma 1 from Anderson & Trapp's Shorted Operators, II isLet $A$ and $B$ be bounded operators on the Hilbert space $\mathcal H$. The following statements are equivalent:
(1) ran($A$) $\subset$ ...
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Problem with Pearson correlation coefficient. [closed]
I have two random variables X and Y. X follows a power law distribution. I know its generating function G(x). I also know the Pearson correlation coefficient of X and Y. How do I find the generating ...
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Is this a well-known probabilistic model?
While I was thinking about the Erdős discrepancy problem, the following random walk model arose rather naturally. You fix a positive integer k, and you take a random step of 1 or -1 at each stage,...
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Conditional probabilities in Banach spaces
This is the infinite-dimensional sequel to my question, Conditional probabilities are measurable functions - when are they continuous?.
Let $\Omega = \Omega_1 \times \Omega_2$ be a probability space ...
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Is there a way to analytically compute the recurrence time of a finite Markov process?
Let $X_t$ be an ergodic (time-homogeneous) Markov process (in discrete or continuous time) on a finite state space $\{1,\dots,n\}$. Let $T(X_0)$ be the stopping time given by the infimum of times such ...
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Infinite electrical networks and possible connections with LERW
I've been exposed to various problems involving infinite circuits but never seen an extensive treatment on the subject. The main problem I am referring to is
Given a lattice L, we turn it into a ...
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Non-existence of integral with respect to Poisson Random Measure
Let $\xi$ be a Poisson Random Measure of intensity $\mu$ (informally $\mathbb E\xi = \mu$).
(For $f \ge 0$, say) when does $\xi f = \infty?$ Kallenberg (Foundations of Modern Probabilility) claims ...
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Equality in the union bound.
Lemma: Let $A_1,\ldots,A_n$ are events $n\in\mathbb{N}$ then
$$
\sum_{i=1}^n \mathbb{P}(A_i) = \mathbb{P}(\cup_{i=1}^n A_i)
$$
if and only if $A_1,\ldots,A_n$ are mutually exclusive.
Both ways are ...
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Correspondence between Viterbi algorithm and Smith-Waterman
Viterbi is an algorithm for finding the maximum likelihood assignment to the hidden variables of an HMM, given the observed variables (we know the transition and emission probabilities of the HMM). ...
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Conditional probabilities are measurable functions - when are they continuous?
Let $\Omega$ be a Banach space; for the sake of this post, we will take $\Omega = {\mathbb R}^2$, but I am more interested in the infinite dimensional setting. Take $\mathcal F$ to be the Borel $\...
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How many trial picks expectedly sufficient to cover a sample space?
Consider a sequence of independent events where an $r$ element subset of an $n$ element set is picked uniformly randomly (ie. any of the $\begin{pmatrix}n\newline r\end{pmatrix}$ possibilities being ...
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Football Squares
Dear Colleagues,
This is a math question for people who know the rules of (American) football.
Every year my barber runs a “football squares” game. He finds 100 customers, each put in 20 dollars, ...
