**69**

votes

**13**answers

16k views

### Statistics for mathematicians

I'm looking for an overview of statistics suitable for the mathematically mature reader: someone familiar with measure theoretic probability at say Billingsley level, but almost completely ignorant of ...

**66**

votes

**8**answers

12k views

### Is there a natural random process that is rigorously known to produce Zipf's law?

Zipf's law is the empirical observation that in many real-life populations of n objects, the $k^{th}$ largest object has size proportional to $1/k$, at least for $k$ significantly smaller than $n$ ...

**30**

votes

**12**answers

11k views

### Why is it so cool to square numbers (in terms of finding the standard deviation)?

When we want to find the standard deviation of $\{1,2,2,3,5\}$ we do
$$\sigma = \sqrt{ {1 \over 5-1} \left( (1-2.6)^2 + (2-2.6)^2 + (2-2.6)^2 + (3-2.6)^2 + (5 - 2.6)^2 \right) } \approx 1.52$$.
Why ...

**28**

votes

**5**answers

2k views

### You pass X people and Y people pass you: how relatively fast are you?

This question occurs to me every time I go jogging. I suspect every runner probabilist in the world must have thought of it (though I'm no probabilist), but I could not specifically find it online. I ...

**26**

votes

**5**answers

4k views

### Inference using Topological Data Analysis: Is it worth it for a regular statistician to learn TDA?

After having read Gunnar Carlsson's Topology and Data I feel enthusiastic to use some topological data analysis (TDA) methods in my current research, mostly in social sciences. We often handle huge ...

**25**

votes

**3**answers

2k views

### Why do statistical randomness tests seem so ad hoc?

Wikipedia describes Kendall and Smith's 1938 statistical randomness tests like this:
The frequency test, was very basic: checking to make sure that there were roughly the same number of 0s, 1s, ...

**21**

votes

**1**answer

5k views

### L1 distance between gaussian measures

L1 distance between gaussian measures: Definition
Let $P_1$ and $P_0$ be two gaussian measures on $\mathbb{R}^p$ with respective "mean,Variance" $m_1,C_1$ and $m_0,C_0$ (I assume matrices have full ...

**21**

votes

**2**answers

1k 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 ...

**19**

votes

**3**answers

1k 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 ...

**18**

votes

**4**answers

793 views

### Applications of algebraic geometry to machine learning

I am interested in applications of algebraic geometry to machine learning. I have found some papers and books, mainly by Bernd Sturmfels on algebraic statistics and machine learning. However, all this ...

**17**

votes

**3**answers

708 views

### Deceptively simple inequality involving expectations of products of functions of just one variable

For a proof to go through in a paper I am writing, I need to prove the following deceptively simple inequality:
$$(*)\qquad E(X^a) E(X^{a+1}\log X) > E(X^{a+1})E(X^a\log X) $$
where $X>e$ has ...

**17**

votes

**2**answers

1k views

### Calculating the “Most Helpful” review

How would you calculate the order of a list of reviews sorted by "Most Helpful" to "Least Helpful"?
Here's an example inspired by product reviews on Amazon:
Say a product has 8 total reviews and ...

**17**

votes

**3**answers

2k views

### What is quantum Brownian motion?

It seems that the current state of quantum Brownian motion is ill-defined. The best survey I can find is this one by László Erdös, but the closest the quantum Brownian motion comes to appearing is in ...

**17**

votes

**1**answer

1k views

### Intuitive Proof of Cramer's Decomposition Theorem

Cramer's decomposition theorem states that if $X$ and $Y$ are independent real random variables and $X+Y$ has normal distribution, then both $X$ and $Y$ are normally distributed. I've seen a few ...

**16**

votes

**5**answers

1k views

### Is a fair lottery possible?

I'm trying to come up with a scheme for a lottery where each individual has roughly the same chance of becoming the winner, regardless of the number of tickets one holds. So no individual should have ...

**16**

votes

**2**answers

2k views

### When is the function of a median closer to the median of the function than the mean of the function is to the function of the mean?

Background
notation: RV= random variable, $\mu=$ mean $m=$ median
Jensen's Inequality considers the relationship between the mean of a function of an RV and the function of the mean of an RV.
If ...

**16**

votes

**1**answer

1k views

### Gini Coefficient and Renyi Entropy

Gini coefficient (aka Gini Index) is a quantity used in economics to describe income inequality. It is 0 for uniformly distributed income, and approaches 1 when all income is in hands of one ...

**15**

votes

**1**answer

572 views

### The Chow & Robbins game ≈ 0.79295350640: improvements could come from simple statistics, or from a continuous version of the game

This question seeks help with improving a numerical estimate of the value of the Chow and Robbins game. Much about this game is unknown, such as whether its value is rational, but there are two routes ...

**14**

votes

**2**answers

424 views

### How to sample uniformly from singular matrices

I would like to uniformly sample from all singular $n$ by $n$ Bernoulli matrices (that is each entry is $1$ or $0$ with probability $1/2$). I could of course just sample from all $n$ by $n$ Bernoulli ...

**13**

votes

**7**answers

3k views

### Correlation and Causation. When can we believe correlation (reasonably, at least) imply causation

We always hear, when reading on correlation, that "correlation does not imply causation."
Still, I have never seen any source that tries to answer the question of when can we reasonably conclude a ...

**13**

votes

**4**answers

1k views

### Are gaussians with different moments far in total variation distance?

If two Gaussians disagree on one moment, it seems like this should imply that they have a large variation distance--equivalently, if two Gaussians are close in variation distance it's hard for their ...

**13**

votes

**2**answers

605 views

### Archaeogenetics

This question is meant to be applied to recover historic information from genetic data.
The following model, is (probably) the simplest possible which takes recombinations into account.
First, let ...

**13**

votes

**0**answers

335 views

### How fast can extreme eigenvalues of the average of random matrices converge to their expectation?

Suppose that $X_1,X_2,\ldots,X_m$ are independent $d\times d$ random matrices and let $\overline{X} := \frac{1}{m}\sum_{i=1}^m X_i$. One of the questions studied under the theory of random matrices is ...

**12**

votes

**3**answers

1k 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 ...

**12**

votes

**4**answers

1k views

### How long for a simple random walk to exceed $\sqrt{T}$?

Let $R_n$ be a simple random walk with $R_0 = 0$, and let $T$ be the smallest index such that $k\sqrt{T} < |R_T|$ for some positive $k$.
What is an expression for the probability distribution of ...

**12**

votes

**1**answer

758 views

### Error to sum of Euler phi-functions

The number theory identity $\phi(1) + \phi(2) + \dots + \phi(n) \approx \frac{3n^2}{\pi^2}$ can be interpreted as counting relatively prime pairs of numbers $0 \leq \{ x,y \} \leq n$ .
Has anyone ...

**12**

votes

**2**answers

2k 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 ...

**12**

votes

**0**answers

426 views

### What kind of random matrices have rapidly decaying singular values?

I've been told that in machine learning it's common to compute the singular value decomposition of matrices in order to throw out all information in the matrix except that corresponding to, say, the ...

**12**

votes

**1**answer

594 views

### Distribution of maximum of random walk conditioned to stay positive

I have an $n$ step random walk which starts at zero $X_0 = 0 = S_0$ where the steps $X_i$ are independent uniform random variates in $[-1,1]$, but the walk is conditioned on the hypothesis that it ...

**11**

votes

**7**answers

1k views

### 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 ...

**11**

votes

**1**answer

646 views

### Applications of the Giry monad in probability and statistics

In another thread, I asked about the $M$ endofunctor on the category $\operatorname{Meas}$ of measurable spaces, which sends a space $X$ to its space of measures $M(X)$.
Will Sawin described the ...

**11**

votes

**2**answers

723 views

### Estimate rate of real correct/wrong from 4 answers quiz.

I recently read that one in ten students think the first man on the moon was Buzz Lightyear, a Toy story cartoon. I'm not here to discuss the data in itself, rather, this reading got me into a problem ...

**11**

votes

**2**answers

526 views

### Covariance of INID order statistics [closed]

In the IID case, it is known that all order statistics are positively correlated.* Thus, we know that $$\text{Cov}(X_{(i)},X_{(j)}) \geq 0.$$ Is this known in the INID (independent, non-identically ...

**10**

votes

**1**answer

2k views

### What's the maximum entropy probability distribution given bounds [a,b] and mean?

What is the continuous probability distribution that maximizes entropy, given only the bounds of the random variable [a,b] and the mean mu of the probability distribution?
For example:
if a=0, b=1, ...

**10**

votes

**3**answers

836 views

### Rapid evaluation of multivariate normal integral

I'm implementing a model that requires me to numerically evaluate a multivariate normal integral of the following form
$$\int_{-\infty}^\infty \phi(z)\displaystyle\prod_{i=1}^N \Phi(a_iz+b_i) \, ...

**10**

votes

**1**answer

250 views

### Normal approximation of tail probability in binomial distribution

My problem: From the Berry--Esseen theorem I know, that $$\sup_{x\in\mathbb R}|P(B_n \le x)-\Phi(x)|=O\left(\frac 1{\sqrt n}\right),$$ where $B_n$ has the standardized binomial distribution and $\Phi$ ...

**10**

votes

**3**answers

335 views

### disconnected or poorly connected graphs in sport ratings systems

I've briefly read about rating systems that provide rankings to players based only on their performance wrt other players, in the context of chess. (for example, elo). When there is a lot of ...

**10**

votes

**1**answer

164 views

### Probability distribution derived from gamma function - does it have a name?

Consider the complex gamma function, denoted by $\Gamma(\sigma+it)$.
Now, let's fix $\sigma$ and let t vary. Then consider the following expression:
$$|\Gamma(\sigma+it)|^2$$
For any choice of ...

**10**

votes

**1**answer

382 views

### (almost) statistical independence of nodes degrees in a graph

Wireless networks are typically modeled as random geometric graphs. The number of nodes $N$ in the network is drawn from a Poisson distribution with intensity $\lambda$
$$P(N = n) = \frac{\lambda^n ...

**10**

votes

**0**answers

837 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 ...

**10**

votes

**0**answers

349 views

### Question from an economist: solving a model of traders' behavior with expectations about the future values of the variable they are currently optimizing

Motivation
I am an economist writing a paper for an academic finance journal. My paper is about the behavior of currency traders, who choose the price at which they will sell currency today, based on ...

**9**

votes

**7**answers

11k views

### Why isn't Likelihood a Probability Density Function?

Hi everyone, first post here...
I've been trying to get my head around why a likelihood isn't a probability density function. My understanding says that for an event X and a model parameter m:
...

**9**

votes

**3**answers

2k views

### Mixtures of Gaussian distributions dense in distributions?

It seems that a mixture of Gaussians can approach any probability distribution, as the number of mixture components approaches infinity. Is this true? And if so, is it precise and correct to say ...

**9**

votes

**2**answers

1k views

### Easier reference for material like Diaconis's “Group representations in probability and statistics”

I'm teaching a class on the representation theory of finite groups at the advanced undergrad level. One of the things I'd like to talk about, or possibly have a student do any independent project on ...

**9**

votes

**7**answers

5k views

### Lower bound for sum of binomial coefficients?

Hi! I'm new here. It would be awesome if someone knows a good answer.
Is there a good lower bound for the tail of sums of binomial coefficients? I'm particularly interested in the simplest case ...

**9**

votes

**3**answers

2k views

### Bayesian statistics for pure mathematicians

Could someone please recommend reading on Bayesian statistics presented from a pure mathematical point of view? That is, works that start assuming a good knowledge of measure theoretic probability. ...

**9**

votes

**2**answers

1k views

### How would you compute that “average” ?

I created a DJ-ing application that allows you to mix your MP3s with a real turntable.
So I generated an audio timecode to burn on a CD, left channel is the absolute position, right channel is a ...

**9**

votes

**2**answers

541 views

### Rescaling positive definite matrices to force a unit eigenvector

Hello,
Let $X'X$ be a positive definite matrix and let $\mathbf{1}$ denote the vector of ones.
I'm hoping to construct a positive, diagonal matrix $W$ such that
$$(W X'X W) \mathbf{1} = ...

**9**

votes

**2**answers

8k views

### Coin Pusher Game

While doing laundry at my local laundromat, I saw a coin pusher game. Below is a picture, and here is a video depicting how it works (disregard non-coins).
Essentially, one has a distribution of ...

**9**

votes

**2**answers

575 views

### Small crown probabilities (and infinite dimensional margin assumption)

My question is:
How do I find sharp upper bounds on $P(|q|\leq \epsilon)$ uniformly over a set of gaussian polynomes $q$ of degree two.
Notations and definitions (to make the question rigorous)
...