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.
9,026 questions
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Area Enclosed by the Convex Hull of a Set of Random Points
Consider $n$ points generated randomly and uniformly on a unit square. What is the expected value of the area (as a function of $n$) enclosed by the convex hull of the set of points?
- 123
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1
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Rigorous proof of the duality of Coupon collector's problem and Occupancy problem
We have $k$ different types of coupons (with replacement).If we collect at least $l$ different coupons, we win a prize. We can only afford to collect $m$ coupons.
Let's say we take all those $m$ ...
- 493
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1
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effective/constructive/algorithmic probability theory
What sort of "alternative" probability theories are out there in which the methods of proof are inherently constructive?
I know of a number of theorems that say that if you take an infinite sequence ...
- 19.7k
6
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1
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Random geometries
Let $M$ be a smooth $n$-dimensional manifold, and let $FM = GL(M)$ indicate its tangent frame bundle. Let $G$ be a fixed linear subgroup of $GL(n)$, and consider the space $\mathcal S$ of all $G$-...
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Sampling when given a set of marginal distributions
There is an unknown joint multivariate distribution P(A_1, A_2, A_3, ..., A_n) (in my scenario, it's a n-dimensional contingency table), which we need to sample from.
Given an arbitrary set of ...
2
votes
1
answer
873
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Bochner's Theorem and Total Positivity
Bochner's Theorem for LCA groups applied to the case of $G = U(1)$ and $G^{\vee} = \mathbb{Z}$ tells us that through the Fourier transform, probability measures on the circle are in bijection with ...
- 952
1
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3
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Random Task Scheduling Problem
Assume there are $m$ tasks, each task's working time conforms to some distribution, for instance an exponential distribution with mean $\lambda$.
So let the r.v. $X_i$ is the working time of the $i$th ...
- 177
1
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0
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131
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Help with derivation of probability density of {event generation} & {event detection}
I would like to specify a new probability distribution that relates to an event of size M being produced by some process and subsequently detected.
Some assumptions :
1) If the event is detected ...
- 11
8
votes
1
answer
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derivative in the Wasserstein space
Villani gives the following formula to find the gradient of a function $F$ of a probability density function $\rho$ in the Wasserstein space :
$$\nabla_W F(\rho) = -\nabla.(\rho \nabla \frac{\delta F}{...
- 481
6
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1
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References for this game
I would like to know how the following game is known in the literature and, possibly, to have references for related papers.
Description of the game: Fix a space $X$ and two Borel probability ...
user11618
8
votes
1
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probability theory for combinatorialists
More than one combinator(ial?)ist has asked me to recommend a good book to learn probability from, and I never know what to say; the probability theory that I use in my research up was mostly learned ...
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Conditional Probabilities - The Mad Kings' Draft
The Problem of the Mad King's Draft:
Suppose there is country which is ruled by a king who can be either 'mad' or 'normal.' The king rules a a large country with a continuum of citizens who have ...
2
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1
answer
530
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Span of Jointly Gaussian Random Variables
Is it true that if $Y_1,Y_2,\cdots,Y_n$ are jointly Gaussian with $E[Y_i]=0~\forall i$ then
$$\mathrm{span}(Y_1,Y_2,\cdots,Y_n) = \mathcal{L}^2(\Omega,\sigma(Y_1,Y_2,\cdots,Y_n),P) ?$$
I wanted to ...
- 23
1
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2
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Stochastic process with Bessel function autocorrelation. (Rayleigh (Jakes) fading for radiowave propagation)
Have the following stochastic process $f(t)$ been studied in mathematics ?
It is stationary, Gaussian, $f(t)-$complex independent Gaussians $N(0,1)$.
The autocorrelation is given by the
zero-order ...
- 24.9k
2
votes
0
answers
240
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$n$-th return of a random walk on $\Bbb Z^d$
Lets define $f_n = P(X_n =0 , X_k \ne 0, k< n)$ the first return distribution of the random walk $X_n$ on $\mathbb{Z}^d$, and lets go ahead and assume that $f_n \approx n^{-(1+\alpha)}$ for some $\...
- 41
4
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2
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Inversion of Moment-generating functions (aka Laplace transform of prob dist)
I want to embark on a project about inverting a Moment-generating function of a probabilitiy distribution. That is given by
\begin{equation}
M_X(t) = \text{E} \exp(tX)
\end{equation}
Since I have ...
- 2,600
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0
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A problem about partial sum of random number composition
Consider the strong random number composition,
$x_1 + x_2 + \cdots + x_n = m$, with $x_i > 0$ and all possible compositions have the same probability.
Let random variable $S_i = \sum_{j=1}^i x_j$...
- 177
1
vote
2
answers
772
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Gibbs sampling step size
I have some data generated using MCMC methods and in particular Gibbs sampling. I computed the autocorrelation but I'm unsure how to determine how many samples to skip.
I'd like to determine that ...
- 87
2
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0
answers
200
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Generator density in $\mathbb{Z}^*_p$
Hello,
Consider the multiplicative group $(\mathbb{Z}/p)^*$ for a given prime $p$. We know that the number of generators in this group is $\phi(p-1)$ --- the Euleur totient function. The question is, ...
- 175
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1
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Reference for difference equations converging to ODE
I am looking for a reference for a result of the following form:
I have a sequence of discrete probability distributions, $p_N$, where the $N$th distribution has associated state space {$k/N, 1 \leq ...
- 1,069
4
votes
1
answer
864
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Maximum entropy probability distribution with known quantile
For continuous distributions on x>0 with known mean m, the exponential distribution f(x) = (1/m)exp(-x/m) is the maximum entropy distribution, with entropy H(f) = ln(m)+1. I have a problem where I ...
5
votes
1
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605
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accumulation points within Pisot numbers
Recall that Pisot numbers are algebraic integers greater than $1$, whose other Galois conjugates have modulus $<1$. The set of Pisot numbers is usually denoted $S$. It is known that $S$ is ...
- 52.3k
1
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1
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Conditional probability and independence
Suppose that we have vectors of events $\{H_1,...,H_n\}$ and $\{D_1,...,D_m\}$. Consider the following two sets of conditions:
Condition set 1
(1) $P(H_i H_j)=0$ for any $i\neq j$ and $\sum_iP(H_i)=...
- 2,619
6
votes
1
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663
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Probability of a set of random vectors over finite field being a spanning set
Suppose I have a set of random vectors $f(a_1, \ldots, a_\ell) := (v_1, \ldots, v_m) \subset F_p^n$, $m \ge n$, given by a matrix valued polynomial function $f$, where the $a_i$'s are independent, ...
- 4,466
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3
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694
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Probability to be the winner in a tournament
In a project in Game Theory we (Ayala Arad and Ariel Rubinstein) are stuck with the following "simple" question. We are sure of the conjecture but we failed to find a (hopefully simple) proof:
Let $...
- 141
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votes
0
answers
81
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Subgraphs of bounded tree-width and preserving edges of original graph
Given a graph $G$, I would like to determine a method for randomly generating subgraphs $G'$ with the following properties:
Each edge of $G$ has at least some probability $p$ of going into $G'$
The ...
- 3,475
4
votes
1
answer
249
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Colored arrangements of circles on the two sphere
Let me define a degree $n$ colored arrangement of circles on $S^2$ to be a collection $\mathcal{C}$ of $n$ disjoint, smoothly embedded circles $C_1,\dotsc, C_n\subset S^2$ together with a ...
- 34.7k
1
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1
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Distance between two points on a one dimensional line
I have a problem similar to the one already described and solved here Mean minimum distance for N random points on a one-dimensional line:
In my case I have N random points on a line of length L ...
- 11
7
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1
answer
394
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Reference request: Martingale decompositions (positive/negative and u.i./singular)
For a paper I am writing, I need these two facts. The proofs are fairly short, but I would rather just cite them. This is for martingales index by natural numbers. Also, I call a martingale which ...
- 6,287
9
votes
2
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726
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Return probabilities for random walks on infinite Schreier graphs
Question: Is there a sequence $(\delta_n)_n$ of real numbers with $\delta_n \to 0$ as $n \to \infty$, such that the following holds:
Let $F$ be a free group on two generators, let $F \curvearrowright ...
- 25.5k
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When does a correlated Brownian motion leave a square?
Let $B=(X,Y)$ be a correlated two-dimensional Brownian motion, that is, the components are standard Brownian motions and the covariance between $X_t$ and $Y_t$ is $t\rho$ for some
constant $\rho \in [-...
- 16.5k
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3
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Is a semicontinuous real function Borel measurable?
Let $f(x,u): [0,1]^2 \mapsto \mathbb{R}$ be a continuous
function.
[Q] Is $g(x) = \inf_{u\in [0,1]} f(x,u)$ always Borel measurable?
If not, can one find a counter-example?
Note that, for any $c$,
...
- 1,399
3
votes
1
answer
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Exchangeable normal distribution mixing measure
I have a zero mean multivariate normal probability distribution where WLOG each marginal variance is unity and all pairwise correlation coefficient are equal and positive. The number of elements in ...
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3
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1
answer
1k
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Divergence between two random variables
I have two Gaussian random variables $X$ and $Y$, each of which is an estimator of an underlying quantity. I need to measure whether $Y$ is estimating something different than $X$. So if the mean of $...
- 33
3
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0
answers
143
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finding rank-3 tensors compatible with a rank-2 tensor projection
I am interested in the following problem: Consider a rank-3 symmetric tensor $\boldsymbol{\sigma}$ with $\sigma_{ijk}$ where $\sigma_{ijk}$ can be 0 or 1, and the symmetry is with respect to any ...
- 41
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votes
1
answer
156
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Expected number of trials to cover certain probability mass for a probability density function?
Suppose we have a univariate random variable $X\sim\mathcal{P}$ with probability density function $f(x):\mathbb{R}\to\mathbb{R}$, $\int_{-\infty}^{\infty}f(x) dx = 1$, we then draw $n$ samples $x_1,...
- 103
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3
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Solving SDE's on subsets of $R^n$.
I posted this on mathstackexchange to no avail.
It is well-known (see for instance Oskendal's text) that if $T>0$ and
$$b(\cdot,\cdot): [0,T] \times \mathbb{R}^n \rightarrow \mathbb{R}^n~~~~~~\...
- 461
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3
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A property of unimodal sequences
It is well-known that $(-1)^j \sum_{i=0}^j (-1)^i\binom{n}{i} \geq 0$. This inequality can be used to prove Bonferroni's inequalities for example. Recently I noticed that a similar inequality applies ...
- 619
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Variance of exponential random variable
For a random variable $\xi$, what bounds can be achieved for Var $e^{\xi}$ in terms of E$\xi$ and Var $\xi$?
- 11
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1
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Mean occurrences of letters in complete strings given by a Bernoulli scheme
Suppose one has an alphabet of $K$ letters, from which we draw sequentially letters; assume that the $n$-th letter occurs with a fixed probability $p_n$ independently of the others and of the previous ...
- 418
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Copulas and marginals thereof
Hello everyone,
I recently became aware of the existence of the copula concept.
So, I have been reading a few things about copulas lately, but
I cannot seem to find information on the following ...
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6
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2
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Given several beta distributions, what is the probability that one is the highest?
Given several random variables distributed according to different beta distributions, how can I calculate the probability that any one of those random variables is actually the highest?
The ...
- 269
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0
answers
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geometric construction of uniform measure on plane partitions in a box
If one picks $a+b$ points uniformly at random (and independently) in $[0,1]$, coloring $a$ of them Red and $b$ of them Blue, then reading the points from left to right one gets a uniform random ...
- 19.7k
5
votes
0
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root system generalizations of Sekiguchi-Debiard (aka Laplace-Beltrami) operators
For the root system $A_n$, taking the limit $q = t^\alpha$ and $t \to 1$, and letting $Y = (t-1) X -1$ one obtains from the Macdonald operator the so-called Sekiguchi-Debiard operator:
$$D_\alpha(X) =...
- 4,466
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2
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884
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A generalization of the Sanov Theorem
Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of i.i.d random variables with law $\mu$. The Sanov Theorem then states that the empirical measures
$$
\mu^N =\frac{1}{N} \sum _{n=1}^N\delta _{X_n}
$$
...
- 2,135
8
votes
2
answers
14k
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Sum of Squares of Normal distributions
Given $X_i \sim \mathcal{N}(\mu_i,\sigma_i^2)$, for $i = 1,\dots,n$. How does one find the distribution of $D = \sum_{i=1}^n X_i^2$? In the case that all the standard deviations are the same (i.e. $\...
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715
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What is the structure of a space of $\sigma$-algebras?
Let $X$ be a compact metric space, and consider the Banach space $\Omega = C(X,\mathbb R)$ of continuous, real-valued functions on $X$, equipped with the supremum norm. Let $\delta_x \in \Omega^*$ be ...
- 8,512
4
votes
2
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835
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Reference on continuous-time finite state filtering
Problem: I'm working in reliability field and have seen papers written on the topic like process of failures when systems are functioning under unobservable (or observable) Markov-like environment, i....
- 267
1
vote
0
answers
336
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Normalized correlation with a constant vector
I am confused how to interpret the result of preforming a normalized correlation with a constant vector. Since you have to divide by the standard devation of both vectors (reference: http://en....
- 171
4
votes
2
answers
515
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Measures that satisfy a 0/1 law
The setting is measure on $2^\omega$. That product (independent) measures obey a 0/1 law, i.e, that measurable tail sets all have measure 0 or 1, is well known. I've made some progress extending this ...
- 153