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Consider a graph, choose some "p: 0<p<1" (probability to infect the neighbor node). Choose some random number "K" of nodes which are "infected" initially. So we can generate epidemic model on a graph …

Background James-Stein estimator and Stein's phenomenon, as described in Wikipedia are rather counterintuitive and amazing. It is claimed that if one wants to estimate the mean $\Theta$ of Gaussian …

Just out of curiosity. The two things sounds a little bit similar - 1) Uncertainty principle 2) Cramer-Rao bound. Saying that we cannot measure something with certain accuracy. However looking closer …

Context Working with some biological datasets it was puzzling to see the patterns like Figure 2 (right) below. The first feeling was, that it corresponds to some biological effects like correlations b …

Consider some manifold $M$ say compact smooth. Let $b_i$ be its Betti numbers (non-zero), i.e. its cohomology dimensions. Assume $M$ can be subsequently fibered by many manifolds, i.e. there is $ M_ …

Let's start from a little bit far. Basic probability theory - chain rule reads: $$ P(AB) = P(A)P(B|A)$$ Example: consider n+m balls, where n - white balls, m - black balls, consider A - first cho …

Standard setup. Consider a group and choose generators. Word-metric (or in the other words - distance on the Cayley graph of the group+generators) - converts a group into a metric space, which is top …

$\newcommand{\Fun}{F_\text{un}}$There was lots of "Fun with $\Fun$" (field with one element) in recent years. One of the points is that it provides bridge between geometrical and combinatorial questi …

Consider $S^k \subset R^{k+1} $. Sample $N$ points by say uniform distribution. (Example k=120, N=2^24, i.e. N>>k ). Consider Voronoi cell around each point. How many neighbours would a cell have …

Consider some compact Riemannian manifold $M$. Fix some point $p$. Consider a "sub-sphere of radius $r$" - i.e. set of points on distance $r$ from $p$. Consider growth function $g(r)$ to be volume of …

Consider any discrete time stochastic process $p(n)$ (price) with independent increments $\xi_k$ and $E(\xi_k)=0$. E.g. Brownian motion (i.e. $\xi_k = N(0,1)$). Consider some "trading strategy" whic …

Consider general Brownian bridge W(0)=0; W(T) = a. (Here "general" means: $W(T)\ne 0$). What is the probability W(t) >= b, for all $ t \in [0, T] $ ? Is there close simple formula in terms of a …

Question: How the maximal height grows for random Tetris like blocks falling down ? Numeric simulation (see below) shows leading term is linear with some constant depending on shapes of blocks allowe …

Setup. Let casino generate a color: black or red with equal probability. Let client try to guess the color. If guess is correct - he earns 1 coin from casino, if not - he gives one to casino. If he lo …

In wonderful lectures by P. Diaconis "Group representations in probability and statistics, Chapter 6. Metrics on Groups, and Their Statistical Use" metrics on permutation groups are considered and cen …