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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Roulette probability [closed]
I'm looking for some knowledge on probability, I've scoured the net but I can't really grasp the answer.
I was having a discussion with a co-worker about roulette probability. He says that at any ...
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Analogues of the Golden-Thompson inequality
Are there any analogues of the Golden-Thompson Inequality for moment generating functions? The Golden-Thompson Inequality asserts the following: If $A$ and $B$ are two $n \times n$ Hermitian matrices, ...
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Reconstructing an ordering of a multiset from its consecutive submultisets
We have a multiset $S$ of size $t$ with $r$ distinct elements, where $t$ is much larger than $r$. We want to reconstruct an ordering $s_1, s_2, ... s_t$ of the elements of $S$ given the values of $t$ ...
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Continuity in intial state of Brownian Motion
$ B = (B_t, \mathcal{F}_t; t\ge 0 ) $ is a 1-d Brownian family on a
measurable space $(\Omega, \mathcal{F})$ with a family of probability
measures $\{\mathbb{P}^x\}$, i.e. $\mathbb{P}^x(B_0
= x) = 1$, ...
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1
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devise a joint distribution of $\alpha$ and $\beta$
If we assume probability density distribution functions of random variables $\alpha$, $\beta$ and $\alpha/ \beta$, we would like to devise a joint distribution of $\alpha$ and $\beta$. Although ...
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2
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Extension of copulas
Let $(X,Y)$ be a random vector. Suppose that the marginal distribution functions of $X$ and $Y$ are known (say $F_1$ and $F_2$). Then the joint law of $(X,Y)$ is given by the following formula:
$$F_{...
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Recovering joint distribution from marginals
Suppose we have a Markov Random Field P(X1,...,Xn) on graph G. Suppose we know P(Xi,Xj) for every edge (i,j). Can we recover P(X1,...,Xn)?
If G is a tree, then there's a formula for joint (product of ...
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approximately linear functions -- more
Suppose $f,g$ are continuous functions from $\mathbb R$ to $\mathbb R$, with the property that
$$f(x)+f(y)=g(x+y)$$
for all $x,y$. Taking $x=y=z/2$ implies that $g(x)=2f(x/2)$ so that the above ...
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Convergence of a series of orthonormal gaussian variables
Does anyone have an idea how to prove the following? It is a step in the proof of some theorem in a book about gaussian processes.
Let $f_n$ be an orthonormal sequence of gaussian variables. Consider ...
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Law of the sum of order statistics through MCMC
Hi everyone,
I was wondering is there were some known applications of MCMC methods for simulating the law of the sum ( or alternatively the joint law) of the order statistics of a multinomial (...
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Estimate probability( 0 is in the convex hull of N random points ) ?
Can anyone estimate N such that Prob( 0 is in the convex hull of $N$ points ) >= .95
for points uniformly scatterered in $[-1,1]^d$, $d = 2, 3, 4, 10$ ?
The application is nearest-neghbour ...
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Looking for a probability distribution
Recently I discussed an experiment with a friend. Assume we start a random experiment. At first there is an array with size 100 000, all set to 0. We calculate at each round a random number modulo 2 ...
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Markov Chain Patterns
Hi
I would like to detect repetitive patterns and deviations from these repetitions. I have historical data and can calculate probabilities for the transitions between my many states. I have ...
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Matrices whose exponential is stochastic
The complex matrix exponential of a Hermitian matrix is unitary: $e^{-iH} = U$. Is there a name or a characterization for matrices Q whose real exponential is stochastic: $e^{-Q} = S$?
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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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univariate prior corresponding to weighted sum of L1 and L2 penalties?
Is there a univariate probability distribution $p_{\lambda,\alpha}(\beta)$ over the reals, parameterized by $\lambda > 0$ and $1 >= \alpha >= 0$, such that $p_{\lambda,\alpha} \propto \exp(-\...
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Independence of conditional hitting distribution and hitting time
Suppose given a d-dimensional Brownian motion $B_t$ starting from the origin and a centered ball with radius 1. Define T as the first hitting time of the sphere (boundary of the ball). How can one ...
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What is the limiting distribution of local minima of n mod i, for i up to $\sqrt{n}$, as $n \rightarrow \infty$?
The sequence n mod i
Consider the sequence n mod i for i=1...$\sqrt{n}$. If we draw the sequence as an xy-plot, we get a dense triangle (since n mod i < i). More precisely, the limiting density of ...
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The probabilistic method - reference to less challenging questions
I am teaching a course in combinatorics and large part of it is dedicated to the probabilistic method especially in the case of graphs. The course is an undergraduate level (almost none of the ...
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Modular Inequality
I'm trying to find a closed form solution to the following probability given two random values $a$ and $b$:
$P(a \mod{p} < b \mod{p}~|~a \mod{q} > b \mod{q},~p \lt q)$
Ideas?
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Assume the standard (better to switch) solution of the Monte Hall problem. Then there's the 3-card Monte problem
Ok, I understand and am convinced by the standard solution of the Monte Hall Problem, i.e. it is better to switch doors after Monte opens one, and improve one's probability of winning from 1/3 to 2/3. ...
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Homology of algebraic varieties in Okounkov's paper on enumerating algebraic curves
This is a series of questions in chronological order. I am lately trying to understand Okounkov's Random surfaces enumerating algebraic curves. So he mentions something about virtual fundamental class....
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Intuitive "proof" or explanation of a result in Friedman's urn
Let $g, r, a, b$ be positive integers. In Friedman's urn model we have an urn with $r$ red and $g$ green balls in it. In each step we take one ball out of urn, register its color and return it to the ...
- 22.3k
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entropy of normal distribution [closed]
What is the entropy of a normal distribution with mean 0 and variance \sigma?
Thanks!
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1
answer
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Does positive density imply existence of the density for some part of a decomposition?
Suppose a $\mathcal{H}^{1}$ measurable set $A\subset \mathbb{R}^{n}$ has positive Hausdorff density $\Theta^{1}(\mathcal{H}^{1},A,x)=c>0$ in a point $x\in A$.
If we have a decomposition $A=B\cup ...
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Expectation maximum i.i.d rv´s
If I have a fixed positive integer $N$ and $N$ i.i.d rv´s. $X_1,X_2,...,X_N$, and parameters $a_i$ such that $\displaystyle\sum_{i=1}^N{a_i}=1$, it is well known that there is a global maximum of
$...
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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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approximately linear functions
i suppose it's fairly well known that if a (continuous, real-valued) function $f$ on the real line satisfies
$f(x-y)=f(x)-f(y)+const$
then it is necessarily linear.
are there any general ...
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Some models for random graphs that I am curious about
G(n,p)
We are familiar with the standard notion of random graphs where you fixed the number n of vertices and choose every edge to belong to the graph with probability 1/2 (or p) independently. This ...
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Does the translation of a random variable preserve its distribution type? [closed]
This might be a very silly question, but I just wanted to make sure I have all the right steps.
Suppose we have a univariate continuous random variable $X$, with some pdf and cdf ${{f}_{X}}(x)$ and $...
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What is Being Counted? [closed]
There are two urns. One contains five white balls. The other contains
four white balls and one black ball. An urn is selected at random and a ball
in that urn is selected at random and removed. ...
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1
answer
257
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Two-Dimensional Gobbling Algorithm
Let (m,n) be an ordered pair of positive integers. While m>0 and n>0, let k_1 be a random positive integer between
1 and m and k_2 a random positive integer between 1 and n. Output (k_1,k_2). Let ...
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Negative Association of Component Size in Random Hypergraph
I have a $d$-uniform hypergraph on $n$ vertices with $k$ hyperedges, where $d << k$ and $n = 4k d^2$ or so.
The hyperedges are placed independently uniformly at random. I would like to have a ...
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3
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Medium-Sized Calculations and Organization
This is not a math question as much as a process question. For the first time in my (very short) career, I find myself doing one of those messy calculations, where each 'line' of the calculation can ...
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2
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258
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near independence of markov chain observations at high lags
I have to simulate independent draws from a very complicated distribution. They only feasible way appears to be using MCMC. I was considering running thousands of chains in parallel, but that would ...
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0
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Loynes spaces, also called pseudo-Hilbert spaces
Let me first define my object:
First, a locally convex space $Z$ is called admissible in the sense of Loynes if
$Z$ is complete
There is a closed convex cone in $Z$, called $Z_+$, satisfying (for $x\...
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answers
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Mathematics of the Anthropic Principle [closed]
A form of the anthropic principle is as follows: "We can observe the universe only because we can exist within it in some way such that we can observe it, and it exists such that we can observe it."
...
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Infinite System of Stochastic Ordinary Differential Equations Coupled by Infinite Graphs
$\ \ \ $ In Time Evolution of Infinite Anharmonic Systems,
Lanford,Lebowitz and Lieb, roughly speaking, proved that for some families of functions $F_v$ $(v\in\mathbb Z^d)$
and a large set of initial ...
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1
answer
100
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Ranking sources at variable(random) frequencies
Hi,
I have this math modeling problem that I need help with. If I have 3 data sources, each being updated at different frequencies, what would be the best way to rank them so the less frequent ...
- 346
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2
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Failure probability formula
In page 21 of A Problem seminar, D. J. Newman presents a novel way (at least for me) to determine the expectation of a discrete random variable. He refers to this expression as the failure probability ...
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3
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An Integral and derived double integral
Suppose that $f\left(x\right)\geq0$ is continuous on $\left[-\infty,\infty\right]$
and $\int_{-\infty}^{\infty}f\left(x\right)dx=1$. Is it true that
$\int_{-\infty}^{\infty}\left|x\right|f\left(x\...
- 3,069
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2
answers
521
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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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1
answer
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edge distribution of random Young's tableaux from Okounkov's "random matrices and random permutations"
I am reading the paper "random matrices and random permutations" by Andrei Okounkov, which is very beautifully written. I just have several technical questions about some of the computations:
1. in ...
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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 ...
CommunityBot
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Feynman Kac Formula as appears in Krzysztof Gawedzki's Lectures on conformal field theory
The lecture notes appeared in the second volume of "Quantum Fields and Strings, a course for mathematicians". I would like to understand the derivation of (1.3), the 2-point correlation function:
$$ \...
- 4,577
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1
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467
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Distribution on permutations derived from probability of pairwise orderings
A followup question to Probability estimates for pairwise majority votes - I think it doesn't actually give an answer in any terribly precise sense, but it would give something I'd be happy to use in ...
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2
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0
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313
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Finding jump probabilities from mean-occupancy values for positions on a one-dimensional random walk
Please imagine a discrete random walk on a one-dimensional lattice. The lattice consists of a set of $L$ positions, $(x_0, x_1, ..., x_L) \in L$, where $x_0$ is the initial position of the walk (as ...
- 599
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3
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423
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probabilities of increasing events under different product measures.
Let $(B_i)$ be a collection of i.i.d. random variables taking values 0 or 1.
Suppose $0 < x^- < x^+ < 1$. Consider two different "success probabilities" for the i.i.d. collection $B_i$: ...
- 3,937
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1
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514
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Shuffling decks of cards where not all cards are distinguishable
Suppose a deck of cards consists of $a_1+a_2+\cdots+a_k$ cards of $k$ types, where there are $a_i$ indistinguishable cards of each type. How many shuffles does it take, on average, to randomize the ...
CommunityBot
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1
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980
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Maximum of Convex combination of random variables
Let $X,Y$ be two independent random normal standard distributions. Consider a function $u(x)=\sqrt[ ]{x}$ if $x\geq{}0$ and $u(x)=-2\sqrt[ ]{-x}$ if $x<0$. Define $Z=aX+(1-a)Y$.
Question: How do ...
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