**3**

votes

**0**answers

148 views

### Maximization of a total variation distance subject to another total variation distance in Markov chain

Suppose two dependent random variables $X$ and $V$ from finite alphabets $\mathcal{V}$ and $\mathcal{X}$ with known joint and marginal distributions are given. Let $P_{XV}$ and $P_X$ and $P_V$ are the ...

**29**

votes

**4**answers

2k 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 ...

**29**

votes

**14**answers

23k 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 ...

**2**

votes

**1**answer

151 views

### Probability a Brownian particle with an exponentially distributed lifetime hits a sphere before vanishing

Imagine I have a point source $p_0 = (x_0,y_0,z_0)$ that releases a point-like Brownian particle with a lifetime given by an exponentially distributed rate parameter $\lambda$. When the particle's ...

**83**

votes

**10**answers

9k 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 ...

**16**

votes

**3**answers

1k 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 ...

**9**

votes

**3**answers

856 views

### Expected second moment for random points on a circle

Let $S$ be a circle with unit circumference. Suppose that $n$ random points are chosen independently uniformly from $S$; choosing one arbitrarily as $x_1$, label the rest $x_2, \dots, x_n$ in ...

**8**

votes

**1**answer

230 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 ...

**7**

votes

**1**answer

487 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 $a=1/4$.
...

**5**

votes

**2**answers

202 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 ...

**4**

votes

**4**answers

1k views

### Mean minimum distance for N random points on a unit square (plane)

A previously posted question "mean minimum distance for N random points on a one-dimensional line" produced an elegant answer: for a line of length L, the expected minimum distance (between random ...

**10**

votes

**10**answers

3k views

### A problem of an infinite number of balls and an urn [closed]

I think that the following problem originated in a probability textbook :
You have a countably infinite supply of numbered balls at your disposal. They are all labeled with the natural numbers ...

**6**

votes

**1**answer

175 views

### The time to drift a binary string from one state to another via deterministic selection of two possible random bit mutation procedures

I have some length $L$ binary string consisting of an ordered array of bits: $(b_1, b_2, ..., b_{L})$, where all bit values $b_i$ are originally set to zero. I'd like a particular set of $n$ bits to ...

**2**

votes

**1**answer

254 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 ...

**55**

votes

**17**answers

34k 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:
...

**72**

votes

**24**answers

19k 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 ...

**15**

votes

**5**answers

3k 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?

**40**

votes

**4**answers

2k 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 ...

**19**

votes

**6**answers

5k 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 ...

**34**

votes

**1**answer

2k views

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

The following problem was brought to my attention by a doctoral dissertation on Mathematics Education, but - as far as I know - the solution remains unknown.
I have already asked this question on ...

**33**

votes

**3**answers

2k views

### the following inequality is true，but I can't prove it

The inequality is
\begin{equation*}
\sum_{k=1}^{2d}\left(1-\frac{1}{2d+2-k}\right)\frac{d^k}{k!}>e^d\left(1-\frac{1}{d}\right)
\end{equation*}
for all integer $d\geq 1$. I use computer to verify ...

**21**

votes

**2**answers

868 views

### Drawing natural numbers without replacement.

Suppose we start with an initial probability distribution on $\mathbb{N}$ that gives positive probability to each $n$. Let's call this random variable $X_1$ so we have $P(X_1=n)=p_{1,n}>0$ for all ...

**20**

votes

**5**answers

2k views

### A random walk with uniformly distributed steps

The following problem has bothered me for a long time.
Let us imagine a point on the real axis. At the beginning, it is located at point $O$. Then it will "walk" on the real axis randomly in the ...

**18**

votes

**6**answers

1k 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 ...

**17**

votes

**1**answer

954 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 ...

**16**

votes

**5**answers

2k views

### Brownian motion and spheres

Consider a Brownian motion on $[0;1]$. A (finite) discrete approximation of this Brownian motion consists of $N$ iid Gaussian random variables $\Delta W_i$ of variance $\frac{1}{N}$:
$$ ...

**30**

votes

**5**answers

4k 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 ...

**28**

votes

**1**answer

2k views

### An Entropy Inequality

Let $X,Y$ be probability measures on $\{1,2,\dots,n\}$, and set $K=\sum_i\sqrt{X(i)Y(i)}$ so that $Z:=\frac{1}{K}\sqrt{XY}$ is also a probability measure on $\{1,2,\dots,n\}$. How can we prove the ...

**26**

votes

**3**answers

952 views

### Random knot on six vertices

This question is inspired by Joseph O'Rourke's beautiful question on random knots. Choose an random ordered 6-tuple of points on the unit sphere in $\mathbf{R}^3$, and form a knot by connecting ...

**16**

votes

**3**answers

991 views

### The Bruss-Yor conjecture about an iterated integral

Is the sequence $$w_n=n! \int_0^{1/2} \int_{x_1}^{2/3} \cdots\int_{x_{n-2}}^{\frac{n-1}{n}} \int_{\frac{n}{n+1}}^1 dx_n dx_{n-1} \cdots dx_1$$ increasing for $n\ge 3$?
This is a conjecture of F. ...

**14**

votes

**2**answers

538 views

### Persistent homology of Gaussian Fields in Euclidean space

If you generate points in $\mathbb R^n$ via a process that respects a Gaussian normal distribution, then compute the persistent homology / barcodes, to my eye something fairly regular seems to be ...

**13**

votes

**1**answer

512 views

### Randomly switching street lights, in a square city

This is a combinatorics-probability question, best stated however in "recreational" terms. Imagine a $N\times N$ city, meaning that we have $N$ horizontal streets, and $N$ vertical streets. At each ...

**9**

votes

**3**answers

6k views

### Maximum of Gaussian Random Variables

Let $x_1,x_2,…,x_n$ be zero mean Gaussian random variables with covariance matrix $\Sigma=(\sigma_{ij})_{1\leq i,j\leq n}$.
Let $m$ be the maximum of the random variables $x_{i}$
$$
...

**7**

votes

**3**answers

1k views

### randomness in nature [closed]

What is the explanation of the apparent randomness of high-level phenomena in nature?
For example the distribution of females vs. males in a population (I am referring to randomness in terms of the ...

**9**

votes

**2**answers

438 views

### When can a freely moving sphere escape from a 'cage' defined by a set of impassible coordinates?

To ask this question in a (hopefully) more direct way:
Please imagine that I take a freely moving ball in 3-space and create a 'cage' around it by defining a set of impassible coordinates, $S_c$ ...

**8**

votes

**5**answers

1k views

### Another colored balls puzzle (part II)

The same colleague as in Another colored balls puzzle asked me the following variant which she called "part II".
Imagine you have $n$ balls in a bag that are colored from $1$ to $n$. At each turn ...

**20**

votes

**2**answers

1k views

### Intuitive proof that the first (n-2) coordinates on a sphere are uniform in a ball

It is a classical fact that if $(x_1,\ldots,x_n)$ is a random vector uniformly distributed on the sphere $S^{n-1} \subseteq \mathbb{R}^n$, then the random vector $(x_1,\ldots,x_{n-2})$ is uniformly ...

**12**

votes

**1**answer

662 views

### Idempotent measures on the free binary system?

Let $(S,*)$ be the free (non associative) binary system on one generator (so $S$ is just the set of terms in $*$ and $1$). There is an extension of $*$ to the space $P(S)$ of finitely additive ...

**11**

votes

**3**answers

401 views

### An “inchworm-like” random walk on an integer interval

Imagine I place $k$ stones on an infinite one-dimensional integer interval $Z$ s.t. no stone is more than some distance $d$ from any other stone. For example, if $d=1$ and $k = 5$, we might place the ...

**10**

votes

**3**answers

587 views

### Probability of coprime polynomials

Given positive integer $N$, we choose $m_1, m_2, n_1, n_2$ independently and with equal probabilities from $\{0,1,\ldots,N\}$, and let
$f_1 = x^{m_1} + (1+x)^{n_1}$ and $f_2 = x^{m_2} + (1+x)^{n_2}$ ...

**2**

votes

**1**answer

3k views

### Conjugate prior of the Dirichlet distribution?

What is the conjugate prior distribution of the Dirichlet distribution?

**13**

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 ...

**12**

votes

**5**answers

2k views

### Proof of Krylov-Bogoliubov Theorem

Where can I find a proof (in English) of the Krylov-Bogoliubov theorem, which states if $X$ is a compact metric space and $T\colon X \to X$ is continuous, then there is a $T$-invariant Borel ...

**12**

votes

**3**answers

2k views

### A Markov process which is not a strong markov process?

Can anyone give an example of a Markov process which is not a strong Markov process? The Markov property and strong Markov property are typically introduced as distinct concepts (for example in ...

**9**

votes

**1**answer

348 views

### Probability that a random distance function is metric

Take a random $n \times n$ nonnegative symmetric matrix $D$ with zero diagonal. What is the probability that it is an abstract distance matrix, i.e. satisfies $D_{xy}+D_{yz} \geq D_{xz}$ for all index ...

**9**

votes

**5**answers

1k views

### Metrization of weak convergence of signed measures

Edit: Changed from "Hausdorff" to "metric" spaces.
Let $\mathcal{M}(\Omega)$ denote the space of signed regular Borel measures on a compact metric space $\Omega$. By Riesz-Markov, this is the dual ...

**9**

votes

**3**answers

1k views

### Area Enclosed by the Convex Hull of a Set of Random Points

Consider $n$ points generated randomly and uniformly on a unit square. What is the expected value of the area (as a function of $n$) enclosed by the convex hull of the set of points?

**8**

votes

**0**answers

429 views

### Full conditional probabilities and versions of AC?

A probability is a finitely additive measure on a boolean algebra with total measure $1$.
A function $P:\scr B \times (\scr B - \{ 0 \})$ is a full conditional probability on $\scr B$ (for a boolean ...

**8**

votes

**0**answers

236 views

### Eigenvalues of permutations of a real matrix: how complex can they be?

This is sort of complementary to this thread. I’ll repeat the definitions here:
For a matrix $M\in GL(n,\mathbb R)$, consider the $n!$ matrices obtained by permutations of the rows (say) of $M$ and ...

**8**

votes

**3**answers

700 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 ...