All Questions
9,497 questions
430
votes
16
answers
65k
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 ...
- 48.8k
231
votes
13
answers
42k
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 ...
- 65.4k
222
votes
0
answers
18k
views
Why do polynomials with coefficients $0,1$ like to have only factors with $0,1$ coefficients?
Conjecture. Let $P(x),Q(x) \in \mathbb{R}[x]$ be two monic polynomials with non-negative coefficients. If $R(x)=P(x)Q(x)$ is $0,1$ polynomial (coefficients only from $\{0,1\}$), then $P(x)$ and $Q(x)$ ...
- 2,272
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 ...
178
votes
8
answers
31k
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 ...
- 4,874
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 ...
106
votes
5
answers
10k
views
integral of a "sin-omial" coefficients=binomial
I find the following averaged-integral amusing and intriguing, to say the least. Is there any proof?
For any pair of integers $n\geq k\geq0$, we have
$$\frac1{\pi}\int_0^{\pi}\frac{\sin^n(x)}{\...
- 43.2k
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 ...
- 1,695
98
votes
17
answers
123k
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:
...
- 1,107
94
votes
1
answer
11k
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 ...
- 23.5k
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 ...
- 14k
91
votes
8
answers
16k
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^\text{th}$ largest object has size proportional to $1/k$, at least for $k$ significantly smaller than $...
- 114k
89
votes
1
answer
21k
views
Is the largest root of a random polynomial more likely to be real than complex?
This question might be hard because it got $35$ upvotes in MSE and also had a $200$ points bounty by Jyrki Lahtonen but it was unanswered. So I am posting it in MO.
The number of real roots of a ...
- 5,460
79
votes
11
answers
21k
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 ...
- 804
75
votes
11
answers
28k
views
Does War have infinite expected length?
My question concerns the (completely deterministic) card game known as War, played by seven-year-olds everywhere, such as my son Horatio, and sometimes also by others, such as their fathers.
The ...
- 236k
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 ...
73
votes
6
answers
25k
views
What is a cumulant really?
A cumulant is defined via the cumulant generating function
$$ g(t)\stackrel{\tiny def}{=} \sum_{n=1}^\infty \kappa_n \frac{t^n}{n!},$$
where
$$
g(t)\stackrel{\tiny def}{=} \log E(e^{tX}).
$$
Cumulants ...
- 22.1k
72
votes
9
answers
30k
views
When are probability distributions completely determined by their moments?
If two different probability distributions have identical moments, are they equal? I suspect not, but I would guess they are "mostly" equal, for example, on everything but a set of measure zero. ...
- 2,649
66
votes
1
answer
7k
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,
...
- 151k
66
votes
4
answers
4k
views
Perron number distribution
A Perron number is a real algebraic integer $\lambda$ that is larger than the absolute value of any of its Galois conjugates. The Perron-Frobenius theorem says that any
non-negative integer matrix $M$ ...
- 25.1k
65
votes
9
answers
12k
views
Polish spaces in probability
Probabilists often work with Polish spaces, though it is not always very clear where this assumption is needed.
Question: What can go wrong when doing probability on non-Polish spaces?
- 651
63
votes
3
answers
7k
views
A roadmap to Hairer's theory for taming infinities
Background
Martin Hairer gave recently some beautiful lectures in Israel on "taming infinities," namely on finding a mathematical theory that supports the highly successful computations from quantum ...
- 24.7k
63
votes
2
answers
4k
views
Guessing each other's coins
I recently thought about the following game (has it been considered before?).
Alice and Bob collaborate. Alice observes a sequence of independent unbiased random bits $(A_n)$, and then chooses an ...
- 4,653
62
votes
7
answers
10k
views
Why is the Gaussian so pervasive in mathematics?
This is a heuristic question that I think was once asked by Serge Lang. The gaussian: $e^{-x^2}$ appears as the fixed point to the Fourier transform, in the punchline to the central limit theorem, as ...
- 631
60
votes
10
answers
14k
views
"Surprising" examples of Markov chains
I am looking for examples of Markov Chains which are surprising in the following sense: a stochastic process $X_1,X_2,...$ which is "natural" but for which the Markov property is not obvious at first ...
- 708
60
votes
4
answers
3k
views
Flipping coins on a budget
A coin is flipped $n$ times and you win if it comes up heads at least $k$ times. The coin is unusual in that you're allowed to pick the probability $p_i$ that it comes up heads on the $i$th flip, ...
- 82.6k
60
votes
1
answer
7k
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-...
- 7,831
58
votes
12
answers
30k
views
Is pi a good random number generator?
Part of what I do is study typical behavior of large combinatorial structures by looking at pseudorandom instances. But many commercially available pseudorandom number generators have known defects, ...
- 19.7k
57
votes
4
answers
15k
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 ...
- 7,857
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 ...
- 5,670
55
votes
5
answers
5k
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 ...
- 7,005
54
votes
4
answers
9k
views
Why is Quantum Field Theory so topological?
I understand that my question suffers from my lack of knowledge about the field, but as a mathematician without much knowledge of physics I have been wondering much about the following and I always ...
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 ...
- 11.4k
52
votes
5
answers
2k
views
Tetris-like falling sticky disks
Suppose unit-radius disks fall vertically from $y=+\infty$,
one by one, and create a random jumble of disks above the $x$-axis.
When a falling disk hits another, it stops and sticks there.
Otherwise, ...
- 151k
52
votes
4
answers
13k
views
Central limit theorem via maximal entropy
Let $\rho(x)$ be a probability density function on $\mathbb{R}$ with prescribed variance $\sigma^2$, so that:
$$\int_\mathbb{R} \rho(x)\, dx = 1$$
and
$$\int_\mathbb{R} x^2 \rho(x), dx = \sigma^2$$
...
- 29.2k
51
votes
3
answers
4k
views
What is the sandpile torsor?
Let G be a finite undirected connected graph. A divisor on G is an element of the free abelian group Div(G) on the vertices of G (or an integer-valued function on the vertices.) Summing over all ...
- 19.2k
51
votes
0
answers
2k
views
Alternating colors on a line: infinitely often or converge?
Suppose we have intervals of alternating color on $\mathbb{R}$ (say, red / blue / red / blue / …). All intervals have independent length, with all red intervals distributed as $\mathbb{P}_{R}$, all ...
- 1,010
49
votes
13
answers
24k
views
Why is it so cool to square numbers (in terms of finding the standard deviation)?
When we want to find the standard deviation of $\{1,2,2,3,5\}$ we do
$$\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 ...
- 673
48
votes
8
answers
5k
views
A sudden smiley? :-)
This is a vague question, and I will no doubt be (properly!) chastised for posing it.
I would like to generate a set $S$ of points in $\mathbb{R}^3$—$|S|$ finite or infinite—which
has the ...
- 151k
48
votes
7
answers
12k
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 ...
- 481
47
votes
7
answers
5k
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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 ...
- 11.4k
46
votes
7
answers
10k
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Conway's game of life for random initial position
What is the behavior of Conway's game of life when the initial position is random? -- We can ask this question on an infinite grid or on an $n$ by $n$ table (planar or on a torus). Specifically ...
- 24.7k
45
votes
5
answers
6k
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Nonstandard analysis in probability theory
I am quite new at nonstandard analysis, and recently I became aware of its use in probability theory mainly through the following two books:
Nelson (1987). Radically Elementary Probability Theory
...
- 1,302
45
votes
1
answer
4k
views
Rolling a random walk on a sphere
A ball rolls down an inclined plane, encountering horizontal obstacles, at which it
rolls left/right with equal probability. There are regularly spaced staggered gaps that let the ball
roll down to ...
- 151k
45
votes
1
answer
6k
views
Anti-concentration bound for permanents of Gaussian matrices?
In a recent paper with Alex Arkhipov on "The Computational Complexity of Linear Optics," we needed to assume a reasonable-sounding probabilistic conjecture: namely, that the permanent of a matrix of i....
- 9,823
44
votes
5
answers
7k
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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 ...
43
votes
8
answers
3k
views
How to quantify noncommutativity?
If I have two operators or finite-dimensional matrices $A$ and $B$, how can I quantify the amount to which they commute or don't commute? (I would consider it a big plus if it is computable easily for ...
- 1,890
42
votes
3
answers
5k
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=...
- 52.3k
41
votes
6
answers
4k
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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 ...
- 151k
41
votes
4
answers
2k
views
What is the probability two random maps on n symbols commute?
It is well known that two randomly chosen permutations of $n$ symbols commute with probability $p_n/n!$ where $p_n$ is the number of partitions of $n$. This is a special case of the fact that in a ...
- 38.6k