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.
9,024 questions
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Draw a Random Line Through a Voronoi Tessellation, What is the Average Number of Voronoi Cell the Line Intersects?
Update: problem reformulation
Following the advice in comments, I now restate my problem using Voronoi
tessellation.
Given a unit hypercube $H_n=\{(x_1,\ldots,x_n)\in \mathbb{R}^n: 0\leq x_i\leq
1\}$...
2
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2
answers
350
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Diffusion processes in probabilistic modelling
I'm working on a PhD project that involves parameter estimation for diffusion processes. I'm based in a machine learning research group, and the emphasis here is strongly on "practical" research.
I'...
3
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3
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Expected Number of Bernoulli trials before you get N more heads than tails
Hello,
I'd like to find the expected number of Bernoulli trials that I'll need before I will get exactly n more heads than tails, given a coin which gets a heads with probability p.
My approach ...
1
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0
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118
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A maximum likelihood -based ranking system
Given a collection of sample rankings, what is the best way to compile them into an aggregate ranking? (Don't worry, I'm working towards a well-defined question.) There are at least two obvious ...
- 3,612
23
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3
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Zeroes of the random Fibonacci sequence
Let $X_n$ be the "random Fibonacci sequence," defined as follows:
$X_0 = 0, X_1 = 1$;
$X_n = \pm X_{n-1} \pm X_{n-2}$, where the signs are chosen by independent 50/50 coinflips.
It is known ...
- 19.2k
1
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1
answer
211
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Max Absolute Difference of Expectations under Change of Measure
Given a sample space $\Omega=\{ 1,\cdots,N \}$, a random variable $x$ defined on $\Omega$ that takes value $x_1,\cdots,x_N$, and a set of strictly positive real numbers $w_1,\cdots,w_N$. Define for ...
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1
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217
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Existence of a special density
Does the following function
$f:\mathcal{P}(\mathbb{N})\rightarrow\{0,1\}$ exist :
$f(\mathbb{N})=1$,
$f(A\cup B)=f(A)+f(B)$ for $A\cap B=\emptyset$,
$f(A)=0$ for finite $A$
- 2,753
2
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3
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453
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name for the distribution of a gamma RV raise to 1/p?
This should be a very easy answer for those who know the distribution. Lately, I am dealing a lot with the following distribution:
$\rho\left(x|u,s,p\right)=\frac{x^{pu-1}p}{s^{u}\Gamma\left(u\right)}...
- 256
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1
answer
219
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Transfer independance from $\mathbb{N}$ to $\mathbb{N}^2$
Hello
Here is a little problem for which I have no clue, and I don't even know if it is difficult.
Does there exist a measurable (!) function $\psi:[0,1]^2\mapsto [0,1]$ such that if $(X_i)_i$ is a ...
- 1,352
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4
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386
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Recovering a function from a set of approximations
We assume that we have a finite set of agents with approximate knowledge about a certain function, and from this collection of approximations we want to recover the actual value of the function.
More ...
- 1,228
8
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2
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Probability of a black path on a random chess board
Take a $2n$ by $2n$ chess board (oriented so the grid lines are vertical and horizontal). Usually there are $2n^2$ squares coloured black and $2n^2$ squares coloured white so that a black square is ...
- 3,217
5
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1
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Numerically finding a Mercer expansion for a given covariance kernel
Let $c(r)$ be a nice, continuous function with compact support. For example, $c(r) = \tfrac 1 5 (1-r)^{11} \big( 5 + 55r + 239 r^2 + 429 r^3 \big)$ for $r \in [0,1]$, and $c(r) = 0$ otherwise.
On ...
- 8,512
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Taylor approximation of a function of a random variable
Suppose we have a random variable $X$ and a smooth function $g$. We want to calculate the expectation value $\mathbb{E}(g(X))$. To be able to write down at least an approximate solution, we perform a ...
- 91
1
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1
answer
560
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Positive martingale representation with jumps
I am looking for a martingale representation theorem for positive semimartingales. Using the answer to this question:
Martingale representation theorem for Levy processes
My best guess is (subject to ...
- 667
2
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1
answer
454
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metric for signal to noise ratio in communication systems
I'm not quite sure about how to define a good measure of the quality of a communication channel with fading and interference. Let us assume the simplest case, where a node in a network receives the ...
- 241
1
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1
answer
570
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Comparing hitting probabilities for two different random walks
Let $p$ be a probability in $]0,1[$, and let $(X^p_i)_{i \geq 1}$ be a i.i.d. family
of variables with law $P(X=1)=p, P(X=-\frac{p}{1-p})=1-p$ (so that $E(X)=0$).
Set $S^p_n=\sum_{k=1}^{n} X^p_k$ for ...
- 3,595
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1
answer
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What are the statistics of prime knots in 3d Random walk?
This question on physics stackexchange https://physics.stackexchange.com/questions/12973/the-entropic-cost-of-tying-knots-in-polymers has a formulation which is perhaps more appropriate for this forum....
- 921
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4
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Statistical computation in matrix. Rows before columns? riddle..
First I'll phrase the question as a riddle, and than as a general math problem.
We have 12 lettered vases $(A,B,...,L)$, in each vase there are 30 numbered balls (1-30). In each ball there is some ...
- 23
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1
answer
752
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Random sampling a symmetric matrix
I'd like to sample the elements of a symmetric square matrix uniformly. For example, for a $N\times N$ matrix, I'd like to only keep $\alpha$% of the matrix elements to build a sparse matrix, while ...
- 481
1
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1
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509
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Proving Uniform Convergence from AS Convergence
I'm working the proof of the Stone-Weierstrass Approximation theorem using probability theory from "A Second Course in Probability" by Ross and Pekoz. The statement of the theorem in the book omits ...
- 13
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0
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230
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Estimating moments of diffusion processes
Hi there,
Suppose I have a diffusion process
$dX_t = a(X_t)dt + b(X_t)dW_t$. Is there a straightforward method for approximating the first few moments of $X_T$ for some time $T$? Clearly, one could ...
- 1,666
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0
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Levy jump measure vs. Levy measure vs. sum of jumps
This question might be a bit basic, but I am struggling to understand the connection between various versions of the Ito's lemma for Levy processes (and semimartingales in general). Could someone ...
- 667
21
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1
answer
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How to compute KL-divergence when PMF contains 0s?
From the Wikipedia page on Kullback-Leibler divergence, the way to compute this metric is to utilize the following formula:
The way I understand this is to compute the PMFs of two given sample sets ...
- 439
2
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1
answer
157
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Inferring the location of a reflecting boundary in a toroidal cage with a Brownian particle
Let's say I have a Brownian particle of some radius $r_b$ and coefficient of diffusion $D$, freely moving about in a toroidal/doughnut-shaped chamber with inner and outer radius $R_{inner}$ and $R_{...
- 599
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0
answers
184
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Integration of discounted normal distribution
Hi
I want to find expectation of integration of normal distribution $\varphi(t)\sim N(0,\sigma\sqrt t)$
but i also want to discount it continuously with parameter $\alpha$.I mean i need to ...
- 1
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2
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Order statistics (e.g., minimum) of infinite collection of chi-square variates?
Hi everyone,
This is my first time here, so please let me know if I can clarify my question in any way (incl. formatting, tags, etc.). (And hopefully I can edit later!) I tried to find references, ...
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Number of lattice points in a random disk of radius r
Consider a disk of radius $r$ centered at $(x,y)$, where $(x,y)$ is chosen from the uniform distribution on $[0,1) \times [0,1)$, and let the random variable $N$ be the number of lattice points in the ...
- 19.7k
2
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0
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153
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Reference request for a result on subsets unlikely to be hit by random walks in a group
Suppose we are performing a random walk in a group. More precisely, we have a finite generating set $S$ of a group $G$ and the probability of walking along generator $s$ is given by $\mu(s)$ for some ...
- 21
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2
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In an Erdős–Rényi random graph, what is the threshold for the property "every edge is contained in at least one triangle"?
Let $G(n,p)$ denote the Erdős–Rényi random graph, where $n$ is the number of nodes and $p$ is the probability for each edge. I'm interested in precisely what range of $p$ the random graph has at least ...
- 7,857
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How far will a random walk on the integers go?
Let $S_n$ be the distance walked at time $n$ in a simple symmetric random walk on $\mathbb{Z}$ (with steps $=\pm1$).
It is well known that $\lim_{n\to\infty}(E(|S_n|)/\sqrt{n})=\sqrt{2 /\pi}$.
Can ...
- 5,103
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1
answer
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Distribution wanted
I have a centered random variables $Y_1,Y_2$ which first 10 moments are given respectively by
$$ 0, 1, 0, 6, 0, 90, 0, 2520, 0, 113400 $$
$$0, 1, 0, 32/3, 0, 36847/100, 0, 436879364/15435, 0, ...
- 1,491
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1
answer
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Wasserstein geometry of measures on manifolds related to the generalized Legendre transform and $d^2/2$-convexity
Let $(M,g)$ be a fixed closed Riemannian manifold, normalized to have volume 1. We'll write $d_M(x,y)$ for the (geodesic) distance between two points $x,y\in M$. I'm interested in the following class ...
- 7,197
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1
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How far can a particle travel from its origin if it exhibits self-avoiding Brownian motion in two-dimensions?
Let's say I have a point-like Brownian particle undergoing two-dimensional diffusion on an infinite plane with the caveat is that the particle can never return to a coordinate that it previously ...
- 599
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1
answer
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Minimum distance distribution between N random points in a cube and the origin
We have $N$ points randomly and uniformly chosen in a cube of side $1$ centered at the origin $O$. This means that the coordinates of the point $P_i$ is a vector of random variables $(X_i,Y_i,Z_i)$ ...
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1
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1
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Combinatorics for a stochastic dynamics problem
Suppose we have a circular arrangement (periodic boundaries) with $M$ sites and we want to distribute $N$ particles over these sites such that there are occupation numbers $n_m$ that respect $\sum_m ...
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1
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When should we expect Tracy-Widom?
The Tracy-Widom law describes, among other things, the fluctuations of maximal eigenvalues of many random large matrix models. Because of its universal character, it obtained his position on the ...
- 2,135
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Probabilistic (and other mathematical) methods of physics without the physics?
Many of the methods of physics are vastly more general than their use in that discipline. For example, information theory overlaps with a lot of statistical mechanics, and the latter actually ...
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Regular Conditional Probability given a natural filtration of a stochastic process
OK, this is kind of re-posting, but I think I can clarify the question more, so it's worth a shot.
Consider a real valued process $(X_t)_{t \leq T}$, cadlag on a probability space $(\Omega, (\mathcal{...
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Entropy of the Ising model
Consider the standard Ising model on $[0,N]^2$ for $N$ large. By that I mean the square-lattice Ising model without external field, inside an $N$-by-$N$ square. What is its entropy for $N$ large? It ...
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Compute the expected value of the next step of a sorted random walk
Here's what I'm thinking about. If you have a random walk (move +1 or -1 at each step) of some fixed length, then if you're at the maximum of the walk, the next step you take is -1 with probability 1. ...
8
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1
answer
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Martingale representation theorem for Levy processes
Is there an equivalent of martingale representation theorem for Levy processes in some form? I believe there is no such theorem in generality, but maybe there are some specific cases?
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3
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Entropy of a measure
Let $\mu$ be a probability measure on a set of $n$ elements and let $p_i$ be the measure of the $i$-th element. Its Shannon entropy is defined by
$$
E(\mu)=-\sum_{i=1}^np_i\log(p_i)
$$
with the ...
- 6,034
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2
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What do we actually know about logarithmic energy ?
In potential theory, the $\textit{logarithmic energy}$ of a Radon measure $\mu$ acting on $\mathbb{C}$ is defined by
$$I(\mu)=\iint\log\frac{1}{|x-y|}\mu(dx)\mu(dy).$$ Of course it is not well ...
- 2,135
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2
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863
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Can the Law of the Iterated Logarithm be strengthened?
http://en.wikipedia.org/wiki/Law_of_the_iterated_logarithm
.$\quad$1. Can the independence assumption be weakened, similar to this?
.$\quad$2. Can the identically distributed assumption be dropped/...
user5810
55
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5
answers
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Random manifolds
In the world of real algebraic geometry there are natural probabilistic questions one can ask: you can make sense of a random hypersurface of degree d in some projective space and ask about its ...
- 7,005
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1
answer
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Probability distributions: The maximum of a pair of iid draws, where the minimum is an order statistic of other minimums?
General question: What is the distribution for the maximum of 2 independent draws from cdf F(x), when we know that the minimum of those same two draws is the kth order statistic of the minimum of n ...
- 233
1
vote
3
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501
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Operator probability in a RPN string
Consider the set $S_n$ of all strings of length $n$ ($n$ integer, $n \geq 3$)
representing an expression in RPN
( http://en.wikipedia.org/wiki/Reverse_Polish_notation. )
Assumptions (to simplify):
...
- 31
11
votes
1
answer
919
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Average over Random Permutations
Consider $S_{n}$ the symmetric group and for each $\sigma\in S_{n}$ let $U_{\sigma}$ be its $n\times n$ permutation matrix. Let $A$ be an Hermitian $n\times n$ matrix. I'm interested in computing the ...
- 3,626
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2
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Is there a corresponding Hahn decomposition theorem for the real-valued Radon measures?
Hello,
As we know that a signed measure $\mu$ on $R$ can be decomposed to the positive part $\mu_+$ and negative one $\mu_-$ by the Hahn decomposition theorem.
My question is whether each real-...
- 1,649
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0
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2k
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Product of correlated random variables
Suppose $X_1,...,X_n$ is a sequence of stationary correlated random variables in $\{-1,+1\}$ such that :
$\mathbb{P}[X_i = +1] = p$,
$\mathbb{P}[X_i = -1] = 1-p$ with $p\in (0,1)$,
and with a ...
- 840