8
votes
2answers
481 views

Rain droplets falling on a table

Suppose you have a circular table of radius $R$. This table has been left outside, and it begins to rain at a constant rate of one droplet per second. The drops, which can be considered points as they ...
1
vote
1answer
208 views

Asymptotic behaviour of a mean

Fix $x>0$ and $c\in\mathbb{N}$. Let $f(x):=\frac{c}{4c-2+2x^2}$ and $$m_N(x):=\frac{1}{N} \sum_{i=0}^{f(x)N} \log(\frac{c N}{2}-i(2c-1))$$ I'm pretty sure $m_N(x)\to\infty$ as $N\to\infty$. I ...
3
votes
1answer
160 views

bounding the probability that a polynomial is near 0

Given a polynomial $p(x_1,\ldots, x_k)$ in $k$ variables with maximum degree $n$, and $x_1,\ldots, x_k \in [0,1]$. Suppose $\max_{x \in [0,1]^k} p(x) = 1$, can we get an upper bound on the probability ...
6
votes
3answers
864 views

Is the function $e^{x^2/2} \Phi(x)$ monotone increasing?

Hello, Here is an interesting problem. It looks elementary, but it has taken me some efforts without solving it. Let $$ h(x) = e^{x^2/2} \Phi(x),\qquad \text{with}\quad \Phi(x):=\int_{-\infty}^x ...
9
votes
2answers
546 views

“Probabilistic ultrafilters?”

A naive question. Let $S$ be a set and let $[0,1]^S$ the set of functions from $S$ to the closed interval $[0,1]$. Suppose given some function $P \colon [0,1]^S \to [0,1]$ satisfying the following ...
16
votes
2answers
1k views

There is mathematics behind the 1989 Tour de France !

The $1989$ Tour was won by Greg Lemond (USA, $1961$ - ), who beat Laurent Fignon (France, $1960$ - $2010$) by $8''$. Yes, eight seconds! The closest tour in history. Let me recall a few rules ...
13
votes
1answer
1k views

Probability of a Point on a Unit Sphere lying within a Cube

Suppose we have a (n-1 dimensional) Unit Sphere centered at the origin: $$ \sum_{i=1}^{n}{x_i}^2 = 1$$ What is the probability that a randomly selected point on the sphere, $ (x_1,x_2,x_3,...,x_n)$, ...
6
votes
1answer
2k views

Big picture concerning Ito integral, Stratonovich integral and standard results in probability theory

I am confused and don't get the big picture concerning the connection between Ito integral Stratonovich integral Standard results in probability theory concerning skewed distributions. Example: ...
3
votes
1answer
773 views

“Nice” Solution to repeated integral

I have a problem wherein I have defined a function $I_r(t) = \int e^{(2r-1)at} \int e^{(2r-3)at} \cdots \int e^{at} dt\cdots dt$, and $I_r(0) = 0$, for $r = 1,2,3,\ldots$. I find that $e^{-ar^2t} ...
3
votes
2answers
274 views

How to fill a simplex with almost disjoint cuboids?

There is an algorithm that give us cuboids in $\mathbb{R}^3$, say $Q_1,Q_2,\ldots$, such that $\cup_{i=1}^{\infty} Q_i$ is the simplex with vertices $(0,0,0), (1,0,0) , (0,1,0), (0,0,1)$, and the ...
7
votes
5answers
751 views

How can I sample uniformly from a surface?

Given an equation of a parametric surface, is there a general way to sample of points uniformly distributed on that surface? I'm interested in this problem for purposes of visualisation - rather than ...