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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 $$\int_\mathbb{R} x^2 \rho(x), dx = \sigma^2$$ Fa …

Let $X$ be a random variable taking values in $\{1,\ldots,n\}$, and let $p_i$ denote the probability of the event $\{X = i\}$. Shannon defined the entropy of $X$ to be the quantity $$H(X) = -\sum_i …

Quite a few probabilistic algorithms for generating random graphs exist in the literature, such as: The Erdős-Rényi model The Stochastic Block model The Watts-Strogatz model The Barabasi-Albert mode …

There is an example of this which is important in machine learning: finding linear maps with the restricted isometry property. Given a set $S$ of $m$ points in $\mathbb{R}^N$ (with $N$ very large) an …

As I asserted in my comments, I think it is too much to hope for a reasonable topology on the space of random variables which makes the map $X \mapsto E(X)$ continuous. This is a bit like hoping that …

Problem: given points $x_1, \ldots, x_n$ in $\mathbb{R}^d$ with $n << d$ and $0 < \epsilon < 1$, find a linear map $T \colon \mathbb{R}^d \to \mathbb{R}^k$, $k << d$, such that $$(1 - \epsilon) ||x_i …

Let $H$ be a separable Hilbert space and let $\{e_i\}$ be an orthonormal basis. My first question is: Is there a probability measure on $B(H)$ such that for $T$ chosen uniformly randomly the matr …

This is cross-posted from math.stackexchange and stats.stackexchange. Probably there is no great answer to this question, but I thought I'd give it a shot here. I'm teaching a class on integration o …

Recall that the Cheeger constant (AKA isoperimetric constant) of a graph $G$ is the infimum of $\frac{\partial S}{vol S}$ over all subsets $S$ of $G$ with volume no larger than $vol(G)/2$. I would li …

One common way to quantify non-commutativity which is especially popular in operator algebra theory and non-commutative geometry is to use the Schatten norms. Given a bounded operator $T$ on a separa …