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191 votes
34 answers
81k 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 ...
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 ...
99 votes
28 answers
14k 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 ...
Erik Davis's user avatar
  • 1,695
91 votes
13 answers
146k 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 ...
Michael Lugo's user avatar
74 votes
16 answers
8k 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 ...
55 votes
16 answers
16k views

Why do we need random variables?

In this MathStackExchange post the question in the title was asked without much outcome, I feel. Edit: As Douglas Zare kindly observes, there is one more answer in MathStackExchange now. I am not ...
Filippo Alberto Edoardo's user avatar
54 votes
4 answers
3k views

When has the Borel-Cantelli heuristic been wrong?

The Borel-Cantelli lemma is very frequently used to give a heuristic for whether or not certain statements in number theory are true. For example, it gives some evidence that there are finitely many ...
Eric Naslund's user avatar
  • 11.4k
23 votes
11 answers
2k views

Random versions of deterministic problems

What are the examples of situations where "randomizing" a problem (or some part of it) and analyzing it using probabilistic techniques yields some insight into its deterministic version? An example ...
19 votes
10 answers
3k views

Probabilistic method used to prove existence theorems

I am aiming for a "big list" of theorems using probability techniques to prove existence of some objects. And in each case, there is an interesting question -- can we find an explicit example? Was the ...
17 votes
13 answers
6k views

Probability in number theory

I am hearing that there are some great applications of probability theory (or more general measure theory) to number theory. Could anyone recommend some good book(s) on that (or other types of ...
13 votes
7 answers
2k views

Finite-space dynamical systems

This question is quite open-ended, but I will formulate several sub-questions that I'll try to make precise. It is about finite-state dynamical system: start with a finite set $X$, with say $n$ ...
Benoît Kloeckner's user avatar
13 votes
2 answers
1k views

A comprehensive list of random walk inequalities?

I am interested in finding a comprehensive list of all noticeable random walk inequalities. ie. $S_n = \sum_{k\leq n} X_i$ for i.i.d symmetric $X_i$ I can only seem to find books/papers that list ...
Xiaomi's user avatar
  • 231
13 votes
4 answers
1k views

What results would follow from or imply "randomness" of the primes?

This question on random versions of deterministic problems reminded me that many conditional results in number theory hold if the primes are in some sense random, and it is common knowledge that the ...
12 votes
12 answers
2k views

What are fun elementary subjects in probability?

I have to read several lectures on probability or applications of probability for high school students (of high level). There is no necessary part I must lecture, that is, my aim is just advertisement....
11 votes
5 answers
4k views

Brownian motion, martingales, Markov Chains - Rosetta Stone

What are the most fundamental/useful/interesting ways in which the concepts of Brownian motion, martingales and markov chains are related? I'm a graduate student doing a crash course in ...
9 votes
1 answer
723 views

Popular mistakes in probability

$\DeclareMathOperator\Var{Var}\DeclareMathOperator\Bern{Bern}\DeclareMathOperator\Pois{Pois}$Question: What not-trivial mistakes do students often make when solving problems in probability theory, ...
8 votes
8 answers
5k views

probability and math puzzle books/references [closed]

Hi All, I'd like to solve some math puzzles, especially in the context of probability theory, but I'm open to other areas too. The kind of problems that does not require much knowledge of mathematics, ...
4 votes
4 answers
1k views

Higher Moments, what are they good for? [closed]

Absolutely nothing? And now seriously - When I studied the basics of probability theory, and even in more advanced topics (random walks, stochastic processes, etc.), I always felt that the mean and ...
Miel Sharf's user avatar
4 votes
1 answer
494 views

Sufficient conditions for a SDE to have a stationary probability measure

Apologies if this question is too basic for MathOverflow. For a smooth Wiener-driven SDE on a non-compact manifold $M$ taking the form $$ dX_t = b(X_t) dt + \sum_{i=1}^k \sigma_i(X_t) \ast dW_t^i $$ ...
Julian Newman's user avatar
4 votes
2 answers
714 views

Mathematical means of studying large and complex but finite systems?

I want a list of the sort of mathematics/mathematical tools that are applied to the study of complex and probabilistic systems in order to make quantitative and qualitative observations about their ...
DoubleJay's user avatar
  • 2,383
4 votes
1 answer
556 views

Symmetries of the standard probability space

The standard probability space $(I, \mathcal B, \lambda)$ consists of the interval $I = [0,1]$, its Borel $\sigma$-algebra $\mathcal B := \mathcal B(I)$ and Lebesgue measure $\lambda$. In applications,...
4 votes
3 answers
286 views

Medium-Sized Calculations and Organization

This is not a math question as much as a process question. For the first time in my (very short) career, I find myself doing one of those messy calculations, where each 'line' of the calculation can ...
frustrated's user avatar
2 votes
9 answers
2k views

Examples of amenable groups other than finite groups

I'm reading about amenable groups. What are explicit examples of nonabelian discrete amenable groups other than finite groups? Perhaps a group presentation or matrix representation would be useful.
2 votes
1 answer
526 views

What are some (popular) references on variants of the classical gambler's ruin problem that exists in literature?

It is fascinating that the gambler's ruin problem which is so ubiquitous in modern probability theory (cf. the Levin-Peres text on Markov chain and Mixing Times) actually dates back to a letter from ...
Aditya Guha Roy's user avatar