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
3
votes
2
answers
921
views
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^...
- 29
11
votes
2
answers
1k
views
Is it a coincidence that the universal parabolic constant shows up in the solution to square point picking?
The expected distance $d$ of randomly selected points within a unit square to the square's center is
$d = \frac{1}{6} P$
where P is the universal parabolic constant
$P = \sqrt{2} + \ln{\left(1+\...
- 1,571
2
votes
2
answers
709
views
Computing equivalent vector of random variables from covariance matrix
Given a covariance matrix, how can I construct a vector of expressions of randomly distributed variables whose covariance matrix is equal to the given one?
EDIT: All variables are normally ...
- 163
11
votes
2
answers
2k
views
Wasserstein distance in R^d from one dimensional marginals
This question occurred to me while I was reading Klartag's papers on central limit theorems for convex bodies.
Given probability measures $\mu$, $\nu$ on (the Borel $\sigma$-field of) $R^d$ with ...
18
votes
1
answer
2k
views
How big is the sum of smallest multinomial coefficients?
Given positive integers $n$ and $d$, let $S$ indicate the list of all $d$-tuples of non-negative integers $(c_1,\ldots,c_d)$ such that $c_1+\cdots+c_d=n$. Let $v_i$ be the value of the multinomial ...
- 2,252
12
votes
2
answers
2k
views
Convex hull of $k$ random points
Suppose we have $k$ realizations of a random variable uniformly distributed over the unit cube $[0,1]^n$.
What is the probability that their convex hull has all of the $k$ points as extreme points?
...
- 413
15
votes
2
answers
755
views
Random noncrossing chords of a circle
Suppose you generate random chords of a circle, with endpoints selected uniformly over the circumference, rejecting any chord that crosses a previously generated chord.
The disk is then partitioned ...
- 151k
18
votes
4
answers
1k
views
Pennies on a carpet problem
I recently read the following "open problem" titled "Pennies on a carpet" in "An Introduction To Probability and Random Processes" by Baclawski and Rota (page viii of book, page 10 of following pdf),...
- 4,952
0
votes
1
answer
241
views
Constraints for different probability measures to have the same expectation.
Take different $D_i \in \mathbb{R} \rightarrow \mathbb{R}$ functions $f_1, f_2$ (i.e. $\exists x : f_1(x) \neq f_2(x)$). We have
$E[f_1(x)] = E[f_2(x)]$
Are there conditions that $f_1, f_2$ must ...
- 151
9
votes
1
answer
756
views
Coalescing random walks: a bound for the full coalescence time?
Start a random walk from each vertex of a graph $G$. Let the walkers evolve independently, except that when two of the walkers meet (ie. occupy the same vertex at the same time), they coalesce into ...
1
vote
1
answer
648
views
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-...
- 113
16
votes
3
answers
791
views
Random products of projections: bounds on convergence rate?
The von Neumann-Halperin [vN,H] theorem shows that iterating a fixed product of projection operators converges to the projector onto the intersection subspace of the individual projectors. A good ...
- 181
113
votes
13
answers
46k
views
What are the big problems in probability theory?
Most branches of mathematics have big, sexy famous open problems. Number theory has the Riemann hypothesis and the Langlands program, among many others. Geometry had the Poincaré conjecture for a long ...
3
votes
2
answers
2k
views
Why is Beta the maximum entropy distribution over Bernoulli's parameter?
Why is Beta(1,1) the maximum entropy distribution over the bias of a coin expressed as a probability given that:
If we express the bias as odds (which is over the support $[0, \infty)$), then Beta-...
- 598
4
votes
2
answers
577
views
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 ...
- 13k
1
vote
3
answers
291
views
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 ...
- 115
1
vote
3
answers
332
views
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 ...
- 221
20
votes
4
answers
870
views
Enumeration and random selection
In Peter J. Cameron's book "Permutation Groups" I found the following quote
It is a slogan of modern enumeration theory that the ability to count a set is closely related to the ability to pick a ...
- 85.6k
12
votes
3
answers
1k
views
Is there a simple inductive procedure for generating labeled trees uniformly at random, without direct recourse to Prüfer sequences?
Suppose you have a labeled tree $T$ on vertices $V=\lbrace 1,\ldots,n\rbrace$ that is drawn uniformly at random from the set of all $n^{n-2}$ such trees. I am seeking an $f$ satisfying the following ...
- 1,068
0
votes
1
answer
551
views
Surface of the cut of an ellipsoid / Marginal density of a multivariate normal over an affine space
So I'm trying to get the marginal density of a multivariate normal over an affine space
if $A$ is a matrix in $\mathbb{R}^p \times \mathbb{R}^n$ for $p < n$ and $B \in \mathbb{R}^n$, $\Sigma$ is a ...
- 1,902
3
votes
0
answers
160
views
Characterizing polyhedron from Brownian particle collisions with a boundary
Please imagine that we have an ordinary 2-sphere, of radius $r_{sphere}$, and some three-dimensional polygon that has all of its points fixed at positions strictly internal to the sphere's surface. ...
- 599
13
votes
4
answers
5k
views
What is known about the Gaussian measure of the unit ball in a Hilbert Space?
Let $X$ be an infinite dimensional separable Hilbert Space with norm $||\cdot||$ and let $\mu$ be a Gaussian measure on $X$ such that $\mu(X) = 1$. What do we know about $\mu(B(0,1))$, where $B(0,1)$ ...
- 617
-3
votes
1
answer
318
views
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 ...
5
votes
0
answers
211
views
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 ...
- 4,922
2
votes
2
answers
739
views
Multinomial transformation for matrices
Suppose we have a vector of probabilities $\mathbf{p}=(p_1,...,p_n)$, where $p_i>0$ for $i=1,...n$ and $\sum p_i=1$. Define new vector $\mathbf{r}=(r_1,...,r_{n-1})$ in a following way:
$r_i=\log(...
- 203
0
votes
1
answer
1k
views
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 (...
- 481
20
votes
1
answer
4k
views
Using Fisher Information to bound KL divergence
Is it possible to use Fisher Information at p to get a useful upper bound on KL(q,p)?
KL(q,p) is known as Kullback-Liebler divergence and is defined for discrete distributions over k outcomes as ...
- 2,252
2
votes
0
answers
163
views
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 ...
- 14.4k
2
votes
1
answer
349
views
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). ...
2
votes
2
answers
351
views
Discrete probability algorithms
I have a probability problem, which I need to simulate in a reasonable amount of time. In its simplified form, I have 30 unfair coins each with a different known probability of being heads. I then ...
- 41
1
vote
2
answers
1k
views
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 ...
- 21
10
votes
1
answer
3k
views
Applications of Banach-Tarski Paradox to Probability Theory?
I was just curious, since the B-T paradox is a measure theoretic result, if there are any consequences of this paradox in probability theory? Also, is there is a way of stating the B-T paradox in the ...
- 137
2
votes
0
answers
548
views
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 ...
- 133
12
votes
2
answers
2k
views
What is the probability a random Turing machine is isomorphic to a DFA?
This is a sort of Chaitin/Omega constant type question, and so I do not expect this probability to be computable to arbitrary precision. However, it is also a very practical thing to know from the ...
- 2,392
7
votes
1
answer
804
views
Random, Linear, Homogeneous Difference Equations and Time Integration Methods for ODEs
Most methods (that I know of) of numerically approximating the solution of ODEs are "general linear methods". For this type of method, the so-called 'linear stability' is examined by applying the ...
- 103
4
votes
2
answers
853
views
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 &...
- 1,444
15
votes
2
answers
3k
views
Bounding sum of multinomial coefficients by highest entropy one
When does the following hold?
$$\sum_{(i_1,\ldots,i_k)\in E}
\frac{n!}{i_1! \ldots i_k!}
\le \exp(n H^*)$$
where $H^*=\max_{(i_1,\ldots,i_k)\in E} -(\frac{i_1}{n}\log \frac{i_1}{n}+\ldots +\frac{...
8
votes
3
answers
602
views
Decimating the infinite grid graph
Let $G$ be the graph whose nodes are the points of
$\mathbb{Z}^d$ in the nonnegative orthant (i.e., all
coordinates are $\ge 0$), with edges connecting each
pair of points separated by unit distance.
...
- 151k
3
votes
1
answer
2k
views
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 ...
- 33
1
vote
2
answers
655
views
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 ...
- 31
9
votes
2
answers
2k
views
Is the infimum of the Ky Fan metric achieved?
Consider the probability space $(\Omega, {\cal B}, \lambda)$ where
$\Omega=(0,1)$, ${\cal B}$ is the Borel sets, and $\lambda$ is Lebesgue measure.
For random variables $W,Z$ on this space, we define ...
user6096
4
votes
3
answers
361
views
State of the Art in Stable limits, embeddings, etc
This is a fairly broad request for references. I've tried a few hours of googling, but the usual process of chasing names and references doesn't seem to be converging on any must-read books or ...
0
votes
3
answers
770
views
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 ...
- 699
19
votes
5
answers
8k
views
What is the probability that two random walkers will meet?
It is a well known result that a random walk on a 2D lattice will return to the origin see Polya's random walk constant. Based on this, it is not a big stretch to conclude that the random walk will ...
- 301
2
votes
2
answers
356
views
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\...
- 2,044
1
vote
2
answers
323
views
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:
...
- 11
2
votes
1
answer
178
views
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 ...
29
votes
3
answers
3k
views
Perron-Frobenius "inverse eigenvalue problem"
The Perron-Frobenius theorem says that the largest eigenvalue of a positive real matrix (all entries positive) is real. Moreover, that eigenvalue has a positive eigenvector, and it is the only ...
- 2,210
4
votes
1
answer
2k
views
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 ...
- 41
11
votes
7
answers
29k
views
Resultant probability distribution when taking the cosine of gaussian distributed variable
I am trying to do a measurement uncertainty calculation. I have a gaussian distributed phase angle (theta) with a mean of 0 and standard deviation of 16.6666 micro radians. The variance is the ...
- 113