# Tagged Questions

140 views

I have recently come accross the star product of copulas, that is if $A$ and $B$ are 2-copulas and $\{C_t\}_{t\in[0,1]}$ is a family of copulas, then $C(x,y,z) = \int_0^y C_t(\frac{\partial}{\partial ... 1answer 183 views ### Solving a SDE with quadratic drift I am wondering whether the following SDE can be solved explicitly? $$d X_t = X_t^2 d t + X_t d B_t$$ where$B_t$is a standard Brownian motion. If not, can we say some thing about the moments of ... 0answers 46 views ### integral against a gaussian density over an increasing space Consider the following Gaussian density over$\mathbb{R}^{2^n}$$$p_n(\underline{x}):=\frac{\exp(-\frac{1}{2n}\langle C_n^{-1}\underline{x},\underline{x}\rangle)}{\sqrt{(2\pi n)^{2^n} \det C_n}}$$ ... 1answer 140 views ### Inequality of Partial Taylor Series Hi, For a given$\theta < 1$, and$N$a positive integer, I am trying to find an$x > 0$(preferably the smallest such$x$) such that the following inequality holds: $$\sum_{k=0}^{N} ... 0answers 188 views ### Whether does the following equation have only one finite zero? Dear MOs, Here is a calculus problem which bored me for sometime. Let a>0 and b<0 be fixed.Define the following function (EDIT: Following the comment by Barry Cipra, you may only consider ... 2answers 464 views ### Rain droplets falling on a table Suppose you have a circular table of radius R. This table has been left outside, and it begins to rain at a constant rate of one droplet per second. The drops, which can be considered points as they ... 1answer 201 views ### Asymptotic behaviour of a mean Fix x>0 and c\in\mathbb{N}. Let f(x):=\frac{c}{4c-2+2x^2} and$$m_N(x):=\frac{1}{N} \sum_{i=0}^{f(x)N} \log(\frac{c N}{2}-i(2c-1))$$I'm pretty sure m_N(x)\to\infty as N\to\infty. I ... 1answer 160 views ### bounding the probability that a polynomial is near 0 Given a polynomial p(x_1,\ldots, x_k) in k variables with maximum degree n, and x_1,\ldots, x_k \in [0,1]. Suppose \max_{x \in [0,1]^k} p(x) = 1, can we get an upper bound on the probability ... 3answers 794 views ### Is the function e^{x^2/2} \Phi(x) monotone increasing? Hello, Here is an interesting problem. It looks elementary, but it has taken me some efforts without solving it. Let$$ h(x) = e^{x^2/2} \Phi(x),\qquad \text{with}\quad \Phi(x):=\int_{-\infty}^x ... 4answers 1k views ### Textbooks to use as reference for standard calculus and probability topics I am currently working on a paper to be submitted to a US journal (addressed primarily to non-mathematiciansâ€™ audience) where I use (1) some standard calculus stuff (e.g. limits, Taylor expansions, ... 2answers 531 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 ... 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 ... 3answers 657 views ### Expectation of a simple function of multivariate gaussians iid rvs I would like to compute analytically the following expected value:$E\left( \frac{X_i^2}{\sum_j \lambda_j^2 X_j^2}\right)$where the$X_i \approx N(0,1)$are iid. It seems an elementary integral, ... 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)$, ... 1answer 2k views ### Big picture concerning Ito integral, Stratonovich integral and standard results in probability theory I am confused and don't get the big picture concerning the connection between Ito integral Stratonovich integral Standard results in probability theory concerning skewed distributions. Example: ... 1answer 767 views ### “Nice” Solution to repeated integral I have a problem wherein I have defined a function$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} ...
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 ...