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
15
votes
6
answers
8k
views
Any sum of 2 dice with equal probability
The question is the following: Can one create two nonidentical loaded 6-sided dice such that when one throws with both dice and sums their values the probability of any sum (from 2 to 12) is the same. ...
- 261
15
votes
3
answers
1k
views
Is $\prod_{i=1}^\infty (1-\frac{1}{2^{(2^i)}})$ transcendental?
Motivation. In a coin game, a player flips all their coins every turn, starting with just one coin. If the coins all land heads then the game stops; otherwise, the number of coins is doubled for the ...
- 48.9k
15
votes
3
answers
4k
views
Non-diagonalizable doubly stochastic matrices
Are there constructive examples of doubly stochastic matrices (whose rows and columns all sum up to $1$ and contain only non-negative entries) that are not diagonalizable?
- 731
15
votes
3
answers
2k
views
Probability that product is a perfect square
The probability a given integer in $[0,n]$ is a square is $\frac1{\sqrt n}$. What is the probability that if you take two integers uniformly then their product is square?
I know the main term is $\...
- 13.9k
15
votes
3
answers
6k
views
Is there a central limit theorem for bounded non identically distributed random variables?
I have a sequence of centered independent random variables $X_i$ that are all
bounded by one in absolute value. They are not identically distributed, though.
I would like to know if the central limit ...
15
votes
3
answers
13k
views
Maximum of two normal random variables
The main purpose of the following question is to get some intuition and deeper understanding why the presented method works which would hopefully help me in trying to adapt it to the setting I am ...
- 311
15
votes
2
answers
1k
views
Sum of independent random variables
We know that the sum of two independent normal random variables is again a normal random variable. But is the reverse right? If $X$ and $Y$ are independent random variables satisfying $X+Y$~$N(\mu,\...
- 153
15
votes
1
answer
1k
views
In how many steps a random walk visits all the elements of a finite group, with a probability 1/2?
This question is a variation of the return to the origin problem.
Let $G$ be the finite group $\mathbb{Z}/n \times \mathbb{Z}/n$ and let the random transformation $T: G \to G$ such that $T(a,b) = (...
15
votes
3
answers
2k
views
Wiener process related counterexample
The Wiener process is defined by the three properties:
1. $W(0) = 0$,
2. $W(t)$ is almost surely continuous, and
3. $W(t)$ has independent increments with $W(t) - W(s) \sim N(0, t-s)$ (for $0 ≤ s &...
- 1,101
15
votes
2
answers
5k
views
What areas of algebra could be interesting to probability theorists?
I would like to find some topic of algebra (beyond linear algebra; algebraic number theory is fine) that would be interesting both to a student that wants to specialize in probability theory and to me ...
- 16.9k
15
votes
1
answer
703
views
Information inequalities
What are the feasible $2^n$-tuples of entropies $h(S) := H(X_{i_1},\dots,X_{i_{|S|}})$ where $X_1,\dots,X_n$ are discrete random variables with some (unknown) joint probability distribution as $S=\{...
- 19.7k
15
votes
5
answers
921
views
What fraction of n x n invertible integer matrices contain at least one unit?
The question is simple:
What fraction of matrices in $G_n = \text{GL}_n(\mathbb{Z})$ have at least one unit entry (i.e., either $\lbrace\pm 1 \rbrace$)?
I'm not sure what the correct measure on $...
- 15.5k
15
votes
2
answers
10k
views
Convergence of moments implies convergence to normal distribution
I have a sequence $\{X_n\}$ of random variables supported on the real line, as well as a normally distributed random variable $X$ (whose mean and variance are known but irrelevant). I know that the ...
- 12.8k
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
15
votes
3
answers
2k
views
entropy and flatness of densities
I was reading C.R Rao's Linear Statistical inference. Rao presents the entropy of a continuous distribution (expectation of -log density) as a measure of closeness to the uniform distribution, and ...
- 549
15
votes
3
answers
1k
views
How to shuffle a deck by parts?
This question is mainly a curiosity, but comes from a practical experience (all players of Race for the galaxy, for example, must have ask themselves the question).
Assume I have a deck of cards that ...
- 14.4k
15
votes
4
answers
1k
views
The critical value of percolation on Cayley graphs.
Let $\Gamma$ be a discrete group with a generating set $S$. Let $p_c(\Gamma,S)$ be the critical probability for percolation of the Cayley graph of $\Gamma$. Is it known that if $\Gamma$ is non-...
- 4,705
15
votes
4
answers
2k
views
Why do Littlewood-Paley projections behave like iid random variables
I have read more than once that the Littlewood-Paley (LP) projections of a function (i.e. decomposing a function into parts with frequency localization in different octaves) behave in some sense like ...
- 153
15
votes
8
answers
3k
views
How Does Random Noise Typically Look?
How does random noise in the digital world typically look?
Suppose you have a memory of n bits, and suppose that a "random noise" hits the memory in such a way that the probability of each bit being ...
- 24.7k
15
votes
2
answers
571
views
Spearing rolling hula hoops
Or: Stabbing rolling disks.
Imagine there are $n$ unit-diameter disks rolling between $x=0$ and $x=d$,
reflecting off either end.
The disk centers start at a random location within $[\frac{1}{2}, d-\...
- 151k
15
votes
2
answers
734
views
On sums of independent random variables in Banach spaces
Let $(\xi_n)_{n\ge 1}$, $(\eta_n)_{n\ge 1}$ be independent mean-zero random variables with values in a Banach space $X$ such that
$$\sum_n\mathbb P(\xi_n\in A)\le\sum_n\mathbb P(\eta_n\in A)$$for any ...
- 4,535
15
votes
1
answer
1k
views
Math journal publishing work related to combinatorics, probability, counting problems etc.?
I'm a high school student. My peer and I have done some work on the Ballot Theorem counting problem and Catalan Numbers. We have come up with a new proof to the Ballot Theorem and we demonstrate the ...
15
votes
3
answers
2k
views
Disintegrations are measurable measures - when are they continuous?
This is a sequel to another question I have asked.
The notion of disintegration is a refinement of conditional probability to spaces which have more structure than abstract probability spaces; ...
- 8,512
15
votes
4
answers
1k
views
Painting $n$ balls from $2n$ balls red, and guessing which ball is red, game
The game
Lucy has $2n$ distinct white colored balls numbered $1$ through $2n$. Lucy picks $n$ different balls in any way Lucy likes, and paint them red. Lucy then giftwrap all the balls so that it is ...
- 215
15
votes
2
answers
2k
views
Intuitive explanation of Dvoretzky's theorem
I am wondering if anyone has an enlightening explanation of why Dvoretzky's theorem (which says that a high-dimensional convex body has an almost round central section) is true -- there are a number ...
- 96.4k
15
votes
2
answers
1k
views
self-avoidance time of random walk
How many steps on average does a simple random walk in the plane take before it visits a vertex it's visited before?
If an exact formula does not exist (as seems likely), then I'm interested in good ...
- 19.7k
15
votes
1
answer
1k
views
Generating Random Young Tableaux: A peculiar probability identity
In the paper by Greene, Nijenhuis and Wilf, an algorithm is proposed for generating uniformly random Young tableaux of shape $\lambda$. The algorithm is to uniformly randomly pick a starting cell, and ...
- 4,952
15
votes
2
answers
547
views
Random graphs in $\mathbb R^2$ (or random rays from $\mathbb Z^2$)
The model:
Suppose that for each lattice point in $\mathbb Z^2$ we pick a random direction uniformly and independently. At time $t=0$ we start drawing rays starting from each lattice point in the ...
- 85.6k
15
votes
1
answer
687
views
Probability that a random element of a group is trivial
Let $G$ be an infinite group with a finite generating set $S$. For $n \geq 1$, let $p_n$ be the probability that a random word in $S \cup S^{-1}$ of length at most $n$ represents the identity. Is it ...
- 153
15
votes
1
answer
604
views
Is the set of the convolutions of two-point measures dense in the set of all measures?
A measure supported in two points is a measure of the form
$$
\mu=\alpha\delta_a+(1-\alpha)\delta_b,
$$
where $a<b$ and $\alpha\in (0,1)$.
The question is:
Given a finite non-negative measure ...
- 321
15
votes
2
answers
3k
views
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
15
votes
3
answers
2k
views
Card game / options pricing / Brownian bridge question
We play a game. I shuffle a deck of cards and start dealing them face up. After any card you can say "stop", at which point I pay you 1 dollar for every red card dealt and you pay me 1 for every black ...
- 823
15
votes
3
answers
3k
views
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
15
votes
3
answers
2k
views
Distribution of the spectrum of large non-negative matrices
This question is related to that of Thurston. However, I am not interested in algebraic integers, and I wish to focus on random matrices instead of random polynomials.
When considering (entrywise) ...
- 52.3k
15
votes
1
answer
746
views
Recurrence relations whose base case is 'at infinity'
I ran across this recurrence relation in
a paper by Medina and Zeilberger [MZ]
(who got it from [CR]):
$$f(h,t) = \max \left( \frac{1}{2} f(h+1,t) + \frac{1}{2} f(h,t+1) ,\frac{h}{h+t} \right) \;.$$
...
- 151k
15
votes
1
answer
1k
views
Has the technique of "sprinkling" been used in studying random matrices?
In 1982, while studying the component sizes of random subgraphs of a hypercube, Ajtai, Komlós, and Szemerédi introduced a technique that came to be known as sprinkling. In this technique, the edges of ...
- 4,922
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{...
15
votes
1
answer
660
views
Which limit to take as a key applied math decision
The Borel-Kolmogorov paradox refers to situations where non-uniqueness in the notion of conditioning on a set of measure zero leads to apparent contradictions. As a formal matter, one requires ...
15
votes
1
answer
2k
views
Ping-pong relief map of a given function z=f(x,y)
I have an idea to design a type of
Galton's Board
to "draw" a relief map of a given two-dimensional function $z=f(x,y)$.
A typical Galton's Board drops, say, ping-pong balls through a series
of evenly ...
- 151k
15
votes
1
answer
1k
views
Table with the most seated customers in Chinese restaurant process
Suppose we have some initial configuration of people seated at some tables. We start taking new customers and seat them following Chinese restaurant process. Is there some known work on finding the ...
- 151
15
votes
2
answers
6k
views
Distribution of inverse of a random matrix
I got stuck into a problem and couldn't find its
satisfactory answer anywhere.
My question is simple. Suppose I have a fat random matrix (i,e., $R$ has dimensions $k\times d$ where $k<d$) whose
...
- 151
15
votes
0
answers
398
views
Will a unit disk be completely covered by randomly placed disks of area $\pi,\frac{\pi}{2},\frac{\pi}{3},\dots$ with probability $1$?
On a "bottom" disk of area $\pi$, we place "top" disks of area $\pi,\frac{\pi}{2},\frac{\pi}{3},\dots$ such that the centre of each top disk is an independent uniformly random ...
- 3,567
15
votes
0
answers
477
views
Quantitative Skorokhod embedding
The Skorokhod embedding theorem says that any random variable $X$ with $\mathbb E X=0$ and $\mathbb E[X^2]<\infty $ can be written as $X=B_{\tau }$ where $B$ is a Brownian motion and $\tau $ is a ...
- 723
15
votes
0
answers
749
views
Prove $\int_{0}^{\infty} \cos(\omega x) \exp(-x^{\alpha}) \, {\rm d} x \ge {\alpha^2 \sqrt{\pi} \over 8} \exp \left( -\frac{\omega^2}{4} \right)$
I would like to prove that
$$\int_{0}^{\infty} \cos(\omega x) \exp(-x^{\alpha}) \, {\rm d} x \ge
{\alpha^2 \sqrt{\pi} \over 8} \exp \left( -\frac{\omega^2}{4} \right)$$
for any $\omega > 0$ and $...
- 178
15
votes
0
answers
477
views
Expanding disks lead to what packing of the plane?
Suppose one sprinkles points uniformly at random on the infinite Euclidean plane,
with some density $\rho$ per unit area.
View the points as disks of radius zero.
Now the radii $r$ of all disks grows ...
- 151k
14
votes
8
answers
3k
views
Relevant mathematics to the recent coronavirus outbreak
I would like to ask about (old* and new) reliable mathematical literature relevant to various mathematical aspects of the recent coronavirus outbreak: In particular, standard statistical/mathematical ...
14
votes
3
answers
8k
views
Analog of Chebyshev's inequality for higher moments
I have a positive random variable $X$ with $E[X] = 1$ and a small number $k$ more moments bounded by constants:
$$E[(X-1)^i] = O(1) \forall i \in \{2, ..., k\}.$$
I'd like to bound the average of $n$...
- 241
14
votes
3
answers
4k
views
How to generate random points in $\ell_p$ balls?
How do I feasibly generate a random sample from an $n$-dimensional $\ell_p$ ball? Specifically, I'm interested in $p=1$ and large $n$. I'm looking for descriptions analogous to the statement for $p=2$:...
- 667
14
votes
5
answers
4k
views
Is there an extension of the Arzela-Ascoli theorem to spaces of discontinuous functions?
The Arzela-Ascoli function basically says that a set of real-valued continuous functions on a compact domain is precompact under the uniform norm if and only if the family is pointwise bounded and ...
- 943
14
votes
2
answers
387
views
What are some useful invariants for distinguishing between random graph models?
Quite a few probabilistic algorithms for generating random graphs exist in the literature, such as:
The Erdős-Rényi model
The Stochastic Block model
The Watts-Strogatz model
The Barabasi-Albert model
...
- 29.2k