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bmbumpus.com | ||
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bartoszmilewski.com
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| | | | | Previously: Topology as a Dietary Choice. Category theory lets us change the focus from individual objects to relationships between them. Since topology is defined using open sets, we'd start by concentrating on relations between sets. One such obvious relation is inclusion. It imposes a categorical structure on the subsets of a given set $latex X$.... | |
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almostsuremath.com
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| | | | | The aim of this post is to motivate the idea of representing probability spaces as states on a commutative algebra. We will consider how this abstract construction relates directly to classical probabilities. In the standard axiomatization of probability theory, due to Kolmogorov, the central construct is a probability space $latex {(\Omega,\mathcal F,{\mathbb P})}&fg=000000$. This consists... | |
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mkatkov.wordpress.com
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| | | | | For probability space $latex (\Omega, \mathcal{F}, \mathbb{P})$ with $latex A \in \mathcal{F}$ the indicator random variable $latex {\bf 1}_A : \Omega \rightarrow \mathbb{R} = \left\{ \begin{array}{cc} 1, & \omega \in A \\ 0, & \omega \notin A \end{array} \right.$ Than expected value of the indicator variable is the probability of the event $latex \omega \in... | |
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djalil.chafai.net
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| | | This post is devoted to certain properties of random permutation matrices. Let \( {\mathfrak{S}_n} \) be the symmetric group, \( {n\geq2} \), and let \( {\mathcal{P}_n} \) be the group of \( {n\times n} \) permutation matrices, obtained by the isomorphism \( {\sigma\in\mathfrak{S}_n\mapsto P_\sigma={(\mathbf{1}_{j=\sigma(i)})}_{1\leq i,j\leq n}} \). We know that \( {\mathcal{P}_n} \) is a subgroup of the orthogonal group... | ||
