**124**

votes

**10**answers

15k views

### Is there an introduction to probability theory from a structuralist/categorical perspective?

The title really is the question, but allow me to explain.
I am a pure mathematician working outside of probability theory, but the concepts and techniques of probability theory (in the sense of ...

**275**

votes

**15**answers

39k views

### Why do roots of polynomials tend to have absolute value close to 1?

While playing around with Mathematica I noticed that most polynomials with real coefficients seem to have most complex zeroes very near the unit circle. For instance, if we plot all the roots of a ...

**41**

votes

**11**answers

54k views

### If you break a stick at two points chosen uniformly, the probability the three resulting sticks form a triangle is 1/4. Is there a nice proof of this?

There is a standard problem in elementary probability that goes as follows. Consider a stick of length 1. Pick two points uniformly at random on the stick, and break the stick at those points. What ...

**42**

votes

**9**answers

4k views

### Geometric / physical / probabilistic interpretations of Riemann zeta(n>1)?

What are some physical, geometric, or probabilistic interpretations of the values of the Riemann zeta function at the positive integers greater than one?
I've found some examples:
1) In MO-Q111339 ...

**43**

votes

**1**answer

3k views

### Probability that a stick randomly broken in five places can form a tetrahedron

Edit (June 2015): Addressing this problem is a brief project report from the Illinois Geometry Lab (University of Illinois at Urbana-Champaign), dated May 2015, that appears here along with a foot-...

**21**

votes

**3**answers

2k views

### Probabilities in a riddle involving axiom of choice

The question is about a modification of the following riddle (you can think about it before reading the answer if you like riddles, but that's not the point of my question):
The Riddle:
We assume ...

**11**

votes

**4**answers

4k views

### Is there a simple way to compute the number of ways to write a positive integer as the sum of three squares?

It's a standard theorem that the number of ways to write a positive integer N as the sum of two squares is given by four times the difference between its number of divisors which are congruent to 1 ...

**8**

votes

**1**answer

392 views

### Isomorphisms between spaces of test functions and sequence spaces

I am in the process of writing some self-contained notes on probability theory in spaces of distributions, for the purposes of statistical mechanics and quantum field theory. Perhaps the simplest ...

**8**

votes

**1**answer

358 views

### Twisted random walks

Suppose the points of two random walks in $\mathbb{R}^2$ are given the
step number (or time) as a third coordinate, so that they become paths in $\mathbb{R}^3$.
Here are several pairs of walks of $n=...

**2**

votes

**1**answer

122 views

### Poisson kernel, $E^{(x, y)}\text{exp}\{i\theta X_t - \theta Y_t\} = e^{i\theta x - \theta y}$

Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W_t = (X_t, Y_t)$. How do I see that for any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, y)}\text{exp}\{...

**5**

votes

**2**answers

265 views

### Brownian motion and hitting a Quadrilateral

I want to compute the hitting probability of a bounded plane by a Brownian motion starting at the origin. In other words, given the coordinates of a quadrilateral A , can we compute $P(T_{A}<\...

**18**

votes

**5**answers

5k views

### What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?

What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?

**25**

votes

**1**answer

2k views

### 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 ...

**20**

votes

**6**answers

6k views

### A balls-and-colours problem

A box contains n balls coloured 1 to n. Each time you pick two balls from the bin - the first ball and the second ball, both uniformly at random and you paint the second ball with the colour of the ...

**22**

votes

**1**answer

714 views

### Two conjectures about zero inner products and dissociated sets

The following problems come from something I worked on (with my coauthors) related to proving a new time lower bound for streaming problems. Having worked on these problems for some time with little ...

**30**

votes

**2**answers

2k views

### The probability for a symmetric matrix to be positive definite

Let me give a reasonable model for the question in the title. In ${\rm Sym}_n({\mathbb R})$, the positive definite matrices form a convex cone $S_n^+$. The probability I have in mind is the ratio $p_n=...

**25**

votes

**6**answers

2k views

### Measures of non-abelian-ness

Let $G$ be a finite non-abelian group of $n$ elements.
I would like a measure that intuitively captures the
extent to which $G$ is non-commutative.
One easy measure is a count of the non-commutative ...

**15**

votes

**2**answers

887 views

### Can one view the Independent Product in Probability categorially?

One can construct a category of probability spaces, but this category has no products. Now probability theory relies strongly on the ability to build independent products, the product measure. In a ...

**37**

votes

**5**answers

5k views

### Heuristically false conjectures

I was very surprised when I first encountered the Mertens conjecture. Define
$$ M(n) = \sum_{k=1}^n \mu(k) $$
The Mertens conjecture was that $|M(n)| < \sqrt{n}$ for $n>1$, in contrast to the ...

**47**

votes

**4**answers

5k views

### Connectivity of the Erdős–Rényi random graph

It is well-known that if $\omega=\omega(n)$ is any function such that $\omega \to \infty$ as $n \to \infty$, and if $p \ge (\log{n}+\omega) / n$ then the Erdős–Rényi random graph $G(n,p)$ is ...

**29**

votes

**2**answers

992 views

### Shortest path through $\sqrt{n}$ points out of $n$

Say I sample $n$ points uniformly at random in the unit square, and then I look for the shortest path through $\sqrt{n}$ of those points (rounding up, say). What happens to the length of this path as ...

**22**

votes

**4**answers

2k views

### Distribution of roots of complex polynomials

I generated random quadratic and cubic polynomials with coefficients in $\mathbb{C}$
uniformly distributed in the unit disk $|z| \le 1$. The distribution of the roots of 10000
of these polynomials are ...

**15**

votes

**3**answers

1k views

### Not-lonely runners

The lonely runner conjecture
has several formulations.
They all involve a number $n$ runners running on a circular track,
each with a different speeds, and the conjecture is that each runner is ...

**12**

votes

**2**answers

556 views

### Error term for renewal function

Consider a sequence of independent uniform $[0,1]$ random variables, and for nonnegative real $t$, let $m(t)$ be the expected number of terms in the first partial sum that exceeds $t$. For instance it'...

**22**

votes

**1**answer

2k views

### Expected halting time for “The 2^n Game” (aka 2048) — with random moves

Recently I encountered an online flash game that features an m-by-m grid and input from the directional pad (up, down, left, right). At any point in the game, the grid contains numbers ('blocks') from ...

**16**

votes

**1**answer

534 views

### Reference request: a conjecture of Rota on positive functions of a random variable

Rota and Shen's On the Combinatorics of Cumulants ends with a conjecture which I'll restate as follows:
Let $p \in \mathbb{R}[x_1, x_2, ...]$ be a polynomial such that, for any sequence $X_1, X_2,...

**9**

votes

**4**answers

2k views

### Another colored balls puzzle

This is a puzzle a colleague asked me recently.
Imagine you have $n$ balls in a bag that are colored from $1$ to $n$. At each turn you take two balls at random out that have different colors and ...

**9**

votes

**1**answer

943 views

### Has anyone seen this series?

I come across the following infinite series.
$$
\sum_{n=1}^{\infty} \frac{t^n}{n!\: n^{a}}, \quad\text{for $t>0$ and $a>0$}.
$$
In particular, I am interested in the case where $a=1/4$.
...

**8**

votes

**1**answer

207 views

### Tighter Caratheodory on the moment curve?

The moment curve is the set of points of the form
$$(t,t^2,t^3,...,t^n) \in R^n$$
Let $M$ be the portion of the moment curve where $t\in [0,1]$, and let $\overline{M}$ be the convex hull of $M$.
...

**6**

votes

**2**answers

575 views

### Birkhoff Ergodic Theorem and Ergodic Decomposition Theorem for Continuous-Time Markov Processes

I have a couple of questions regarding ergodicity for Markov processes in continuous time. (In particular, the first question seems like it should be particularly basic, and yet I haven't managed to ...

**5**

votes

**3**answers

1k views

### Packing density of randomly deposited circles on a plane

Let's say that I have a rectangular two-dimensional surface of bounded dimensions, $[0,A]$ and $[0,B]$:
Under "no overlap" constraints, I sequentially deposit circles of radii $r_c$ on this surface,...

**2**

votes

**1**answer

1k views

### Asymptotic behavior of max of chi-squared distribution

Suppose $X_{\max}$ is the maximum in a sequence $X_1,X_2,\ldots,X_n$ where each $X_i\sim\chi^2_k$ is an i.i.d. chi-squared random variable with $k$ degrees of freedom.
Since chi squared distribution ...

**8**

votes

**1**answer

315 views

### Transitive closure of balanced mass transport in Z (move to close)

Given two atomic measures $\mu$ and $\nu$ on $\mathbb{Z}$, write $\mu \sim \nu$ iff there exist countable decompositions $\mu = \mu_1 + \mu_2 + \cdots$ and $\nu = \nu_1 + \nu_2 + \cdots$ along with ...

**4**

votes

**2**answers

252 views

### Anticoncentration of the convolution of two characteristic functions

Edit: This is a question related to my other post, stated in a much more concrete way I think.
I am interested in anything (ideas, references) related to the following problem:
Suppose that $A \...

**3**

votes

**0**answers

637 views

### Has the Lie group preserving a probability distribution been used in Bayesian statistics?

For a (possibly signed) nondegenerate probability measure $\pi$ on $\{1,\dots,n\}$ define
$$\langle \pi \rangle := \{R \in \operatorname{STO}(n): \pi R = \pi \}.$$
Here $\operatorname{STO}(n)$ denotes ...

**7**

votes

**1**answer

442 views

### Location of maximum of Brownian motion with rough drift

I am interested in the distribution of the $\text{argmax}_{t \in [0,1]} \{B(t) + f(t)\}$, where $B$ is a Brownian motion (or Brownian bridge) and $f:[0,1] \to \mathbb{R}$ is a continuous function. ...

**5**

votes

**2**answers

146 views

### Moment matching: construction of a mixture of Gaussian distribution with lower moments identical to Gaussian

This is a question related to the statistical model behind independent component analysis (ICA).
We assume that $Z \sim N(0,1)$. Our goal is to construct a random variable $X$ that follows a ...

**3**

votes

**1**answer

375 views

### Quotients of standard Borel spaces

Let $X$ and $Y$ be standard Borel spaces: topological spaces homeomorphic to Borel subsets of complete metric spaces. Given a surjective Borel map $f:X\to Y$, we get an equivalence relation $\sim_f\...

**107**

votes

**26**answers

34k views

### What is convolution intuitively?

If random variable $X$ has a probability distribution of $f(x)$ and random variable $Y$ has a probability distribution $g(x)$ then $(f*g)(x)$, the convolution of $f$ and $g$, is the probability ...

**84**

votes

**6**answers

10k views

### Why do probabilists take random variables to be Borel (and not Lebesgue) measurable?

I've been studying a bit of probability theory lately and noticed that there seems to be a universal agreement that random variables should be defined as Borel measurable functions on the probability ...

**66**

votes

**12**answers

22k 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 ...

**62**

votes

**21**answers

9k views

### Probabilistic Proofs of Analytic Facts

What are some interesting examples of probabilistic reasoning to establish results that would traditionally be considered analysis? What I mean by "probabilistic reasoning" is that the approach should ...

**65**

votes

**17**answers

61k views

### Google question: In a country in which people only want boys [closed]

Hi all!
Google published recently questions that are asked to candidates on interviews. One of them caused very very hot debates in our company and we're unsure where the truth is. The question is:
...

**58**

votes

**10**answers

11k views

### How is it that you can guess if one of a pair of random numbers is larger with probability > 1/2?

My apologies if this is too elementary, but it's been years since I heard of this paradox and I've never heard a satisfactory explanation. I've already tried it on my fair share of math Ph.D.'s, and ...

**58**

votes

**1**answer

5k views

### The mathematical theory of Feynman integrals

It is well known that Feynman integrals are one of the tools that physicists have and mathematicians haven't, sadly.
Arguably, they are the most important such tool. Briefly, the question I'd like to ...

**47**

votes

**5**answers

3k views

### 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 ...

**51**

votes

**1**answer

4k views

### Why can't a nonabelian group be 75% abelian?

This question asks for intuition, not a proof.
An earlier question,
Measures of non-abelian-ness
was thoroughly answered by Arturo Magidin.
A paper by Gustafson1
proves that, for a nonabelian group,
...

**29**

votes

**7**answers

6k views

### What's the use of a complete measure?

A complete measure space is one in which any subset of a measure-zero set is measurable.
For what reasons would I want a complete measure space? The only reason I can think of is in the context of ...

**23**

votes

**3**answers

1k views

### Some models for random graphs that I am curious about

G(n,p)
We are familiar with the standard notion of random graphs where you fixed the number n of vertices and choose every edge to belong to the graph with probability 1/2 (or p) independently. This ...

**17**

votes

**3**answers

1k views

### “Entropy” proof of Brunn-Minkowski Inequality?

I read in an information theory textbook the Brunn-Minkowski inequality follows from the Entropy Power inequality.
The first one says that if $A,B$ are convex polygons in $\mathbb{R}^d$, then
$$ m(...