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.
9,023 questions
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Concentration results for non-standard Gaussian random vectors.
Given a $c$-Lipschitz function $f(X):\mathbb{R}^d \rightarrow \mathbb{R}$, and given that $X \in \mathbb{R}^d$ is a Gausssian random vector centered at $\mathbb{w} \in \mathbb{R}^d$ (not at zero) ...
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kalman filter: understanding the mathematical part
i am currently reading the Probabilistic robotics book where the filters are discussed.
Such filters as kalman filter or particle filters.
Now I can understand one thing while reading about the ...
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5
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Recommended book for introduction to Chaotic dynamics? (application in probability distributions)
I'm just starting some research and I need a good introductory book in the topic of chaotic dynamics. Does anyone have a suggestion? Thanks.
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Why doesn't Stein effect happen for multinomial distributions?
(Medeen, et all, 1998)" show that Maximum Likelihood estimate is admissible for multinomial distribution under squared error. On other hand, James and Stein showed that arithmetic average is not an ...
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Random linear recurrence relations
Problem
I am interested in the random recurrence relation of the form $x_{n+1}=\alpha x_n \pm \beta x_{n-1}$ where $\alpha$, $\beta$ are known constants and the $\pm$ sign is chosen with equal ...
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Approximation of the law of a stochastic process
Hello Dear fellows,
I thank you in advance for your help and ideas.
I have just read an article and want you to help me understand the rational behind a part of it.
We have two processes $v_t$ and $...
CommunityBot
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How to estimate the growth of the probability that $G(n, M)$ contains a $k$-clique
Let $k\geq 3$ be a fixed positive integer. Define
$t_k(M)=\Pr[G(n, M) \text{contains a}\ k-\text{clique}]$, where $G(n, M)$ is the random graph uniformly distributed on all $n$-vertex graphs with $m$...
CommunityBot
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2
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Elo Rating System Help with the Maths around number of matches
I'm creating a system that will allow people to rate images.
My idea is to use an Elo Rating system (http://en.wikipedia.org/wiki/Elo_rating_system) for each image and then use crowdsourcing to have ...
- 2,401
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351
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Distribution of transformed multinomial variable?
Suppose we have a uniform multinomial distribution over $2^d$ outcomes. Multinomial coefficients give distribution of vector valued variable $v$ where $v$ is the vector of observed counts.
Is there a ...
- 2,252
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5
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Random walk on a two-dimensional uniform grid
Hi,
consider the following random walk on the lattice $\{0,\dots,n\}^2$. It starts at $(0,0)$ and then move either up or right, with probability respectively $p$ and $1-p$. Once it reaches the right ...
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Characteristic operator
Let $X_t\in\mathbb{R}$ be an Ito diffusion process given by $$ dX_t=a(b-X_t)dt+\sigma dW_t$$, then the characteristic operator of $X_t$ is given by $$L=a(b-x)\frac{\partial}{\partial x}+\frac{\sigma^...
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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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2
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573
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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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Why do I not use post hoc tests with a 2 x 2 factorial?
I know this is an obvious answer. I am probably over thinking what I'm doing, but I cannot recall. Does it have to do with not having enough variables to compare the various means?
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Is the nearest walk to Brownian motion uniform?
Let $W:[0,1]\rightarrow\mathbb R$ be standard Brownian motion with $W(0)=0$.
Let $F_n$ denote the collection of all the $2^n$ many piecewise linear continuous functions $f:[0,1]\rightarrow\mathbb R$ ...
CommunityBot
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Is every probability space a factor space of the Haar Measure on some group?
Let P be an arbitrary probability space.
I would like to find a compact topological group $G$ so that the Haar probability measure on $G$ admits a measurable map to the probability space $P$.
By a ...
- 5,898
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211
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Exponential tails for a functional of a subcritical branching process.
Let $(m_i, i \in \mathbb{N})$ be positive weights with $\sum_{i \in \mathbb{N}} m_i^2 < 0.1$.
Consider a subcritical branching process in discrete time and continuous space,
started from some ...
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2
answers
577
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What does the probabilistic model suggest the error term in the PNT should be?
Let $\Lambda(n)$ be the von Mangoldt function. The prime number theorem is equivalent to the statement that $\sum_{n \leq N} \Lambda(n) \approx N$. Defining $\lambda_{*}(n)= \Lambda(n)-1$ we may ...
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Markov chain convergence problem.
Consider a markov chain matrix P of size n x n (n states).
P is known to be:
1- there are at least two absorbent states. one of them is denoted by null. (thus, we have that P_null,null = 1)
2- For ...
CommunityBot
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1
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485
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Looking for a version of Itô's Lemma
Hi Everyone
I have some difficulties deriving the Stochastic Differential Equation for the following problem, any help or reference would be appreciated.
We are given a Brownian Motion $B_t$ and ...
CommunityBot
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2
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0
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530
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About generalization of stirling numbers of the second kind
Hello,
The Stirling numbers of the second kind count how many ways can a set of $k$ elements be partitioned into $n$ non-empty classes, with $k=n,n+1,\dots$.
My question is: Is there a ...
0
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1
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The density of x_1^n+x_2^n where x_i are Gaussian
We define a process $\chi_k^n=\sum _{i=1}^k x_i^n$ where x_i are iid gaussian processes.
I try to find the distribution of $\chi_k^n$. If k=1 then we get $f(x^n=y)=\frac1n y^{\frac{1-n}{n}}\exp(-y^{2/...
CommunityBot
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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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1
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Geneology of survivors in a critical discrete Galton-Watson process
Hello. After flipping through a few textbooks on birth-death processes, I can't seem to find anything about genealogical distribution of survivors (conditioned on non-extinction). What I am looking ...
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1
answer
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Conditional Covariance
It is well known that for two increasing functions $f$,$g$ and for any random variable $X$ then $cov(f(X),g(X))\geq{}0$. Now assume $f,g$ have the same domain $D$ and let $A\subset{}D$. What can I say ...
CommunityBot
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exactness of the Gauss transformation
Dear all,
I would like to know if the Gauss transformation T(x) = fractional part of 1/x, x in (0,1) (with the Gauss invariant probability measure) is an exact endomorphism (in the sense of Rokhlin). ...
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Random geometric graphs and spanners
I would grateful to learn of work mixing
random geometric graphs with random graphs under
the
Erdős-Renyi model, and in particular concerning spanners.
Select $n$ points uniformly at random from the ...
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2
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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 ...
CommunityBot
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648
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Lower bound on the convergence rate of a specific Markov chain
I have a Markov chain $\mathbf{A} = (A_0, A_1, \ldots)$ with state space $\{0, \ldots, n\}$ which converges towards a stationary distribution $\pi$. There are a lot of well-known results on upper-...
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2
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diameter of a graph with random edge weights
Given a weighted directed graph $G=(V,E, w)$, suppose we generate a new graph $G'=(V,E,w')$ with the same vertices and edges, but now letting the weight of edge $(i,j)$ be
an exponential random ...
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1
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3
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Is any bias introduced from initial clustering
I hope this is an appropriate forum for this question, and I asked on math.stackexchange as well. If it doesn't belong, I don't mind closing this. If my questions is not clear, please just let me ...
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3
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Is ERNIE output skewed by statistical tests?
ERNIE is a hardware random number generator used to generate winning Premium Bond numbers in the UK. Wikipedia says: "ERNIE's output is independently tested each month by an independent actuary ...
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2
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very simple conditional probability question [closed]
I know this isn't a research question, so it might get voted off, but here goes:
You know that a couple has two children. You go to the couple's house and one of their children, a young boy, opens ...
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1
answer
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Kernel width in Kernel density estimation
Hi,
I am doing some Kernel density estimation, with a weighted points set (ie., each sample has a weight which is not necessary one), in N dimensions.
Also, these samples are just in a metric space (...
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1
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Porbability of selecting balls from boxes [closed]
There are three boxes. B1, B2, B3 The probability of selecting them is 0.2, 0.2 , 0.6 respectively.
B1 contains 3 red balls and 7 green balls.
B2 contains 5 red balls and 5 green balls.
B3 contains ...
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2
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655
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Expected position of a card in a deck after repeating a procedure
Suppose you have a deck of 52 playing cards, fully shuffled, one of which is the King of Hearts. You perform this procedure:
1. Put the top three cards of the deck into a second pile. If there aren't ...
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How to partially uniformize a deck by partial shuffles?
This question is a variant of a previous one; it was originally a posed as an edit of this former question, but I came to think it could be more suitable to pose it anew.
Assume I have a deck of ...
CommunityBot
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Question regarding divergence
Let $E$ be a closed and convex set of distributions on a finite set $A$. Let $P',Q'\notin E$ and let $P^{\star},Q^{\star}$ be their respective estimates in $E$ with respect to the KL-divergence, i.e.,...
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What will be the distribution of harmonic mean of two correlated gamma random variables?
Suppose there are two correlated random variables $X_1$ and $X_2$ both are gamma distributed but having different shape and scale parameters with correlation coefficient $\rho$. What will be the ...
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2
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Can you explain the description of the Lovasz Local Lemma by Moser+Tardos?
The Lovász Local Lemma (or LLL) concerns itself with the probability of avoiding a collection of "bad" events A, given that the set of events is "nearly independent" (each bad event A &...
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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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1
answer
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sum of order statistics
Suppose I have N real random variables with identical PDF. At every instance of these r.vs, I pick $K$ largest out of $N$. Lets call their sum as $S_K$. Alternatively, based on some criteria, I ...
- 3,937
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2
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356
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Coefficients of holomorphic functions defined by Borel probability measures on the unit disc
Let be $\mathcal M(\partial\mathbb D)$ denote the set of all Borel complex probability measures on $\partial\mathbb D$ (unit circle in the complex plane). Define a mapping $\Phi:\mathcal M(\partial\...
- 1,990
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3
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770
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Skewing the distribution of random values over a range
The following code comment comes from PHP, a free and open source project. I have done my own research and I cannot find any evidence to support the argument made in this code comment. Thus the ...
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2
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323
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Correlation in graph coloring
Let $G$ be a (simple) graph.
Given $k \ge \chi(G)$, define $Cor(G,k,u,v)$ to be the proportion among all $k$-colorings of $G$ for which the vertices $u$ and $v$ have the same color.
Questions:
...
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2
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1
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Maximal inequality over two indices
In Freedman's series of 3 books on Markov processes, I find that I keep on running into terms like:
P[$\max_{0 \leq s \leq 1, s \leq t \leq rs}$ | B(t) - B(s) | > $\epsilon$]
in the background of ...
4
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1
answer
2k
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Covariance of points distributed in a n-ball
Is there a closed form expression for the covariance of a uniform distribution in a n-ball? I would like to develop a test for vector sums of points sampled from a uniform distribution in a n-ball. I ...
- 11.4k
2
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2
answers
487
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Cover time of weighted graphs
Consider a connected graph $G$ with non-negative weights on each edge. The sum of edge weights is the same for each vertex, call this sum $W$. A random walk on the graph at vertex $u$ transitions an ...
- 2,210
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3
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164
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Transforming to uniform numbers
Hi
I have a time series of probabilites, vector X
I need to convert the probabilites to uniform numbers.
As I understand it if I put the series into the cdf the output is thus uniform.
The problem ...
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3
votes
1
answer
539
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Probability of generating the symmetric group
The statement is simple:
What is the probability that a set of n-1 transpositions generates the symmetric group, $S_n$?
The motivation is that I remembered reading that this was an open problem ...
- 17.9k