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Let $\mathcal X$ be either a subset of $\mathbb R$ equipped with Lebesgue measure or a countable set with counting measure. The Gibbs' principle in statistical physics asserts that if $(X_1 , \dots, X …

Let $A$ be a set of $n$ elements. Let $S_1,\dots,S_n$ be independent $k$-element random subsets of $A$. What is the probability that $S_1,\dots, S_n$ evenly cover $A$, i.e. each element of $A$ belongs …

To generate a uniform distribution on a sphere $S^n$ in $\mathbb R^{n+1}$, we can normalize a vector whose entries are $n+1$ i.i.d normal random variables. If $\rho$ is a correlation, $|\rho|<1$, how …

Let $A_0$ be a list of $ n$ distinct elements. By sampling with replacement the elements of $A_0$, we obtain a new list $A_1$ of $n$ elements that are not necessarily distinct. Repeat the same process …

It is known that $$ \cos(\frac{x}{2})\cos(\frac{x}{4})\cos(\frac{x}{8})\dots = \frac{\sin x}{x} = O_{x \rightarrow \infty}(x^{-1}) $$ Is it true that $$ f(x) = \cos(\frac{x}{3})\cos(\frac{x}{9})\cos( …

Let $u$ be taken uniformly from the unit sphere $\mathbb S^{n-1}$ and $D$ be a diagonal matrix. I'd like to find a general formula for $$ \mathbb E[(u^\top D u)^m] $$ for $m=1,2,3, \dots$, in terms of …

Many information theoretic quantities such as entropy and relative entropy appear in rate functions in large deviation theory (LDT). Is there any result in LDT that relates mutual information and rate …

Let $I(x) = \frac{1}{2\pi} \int_{-2}^2 \sqrt{4-y^2}\ln|x-y|dy$. Then $I(x)$ is a concave function and \begin{equation} I(x)= \begin{cases} \frac{1}{4}x^2-\frac{1}{2}, &\text{if } |x|\leq2 \\ \frac{1} …