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mkatkov.wordpress.com | ||
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jmanton.wordpress.com
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| | | | | If $latex Y$ is a $latex \sigma(X)$-measurable random variable then there exists a Borel-measurable function $latex f \colon \mathbb{R} \rightarrow \mathbb{R}$ such that $latex Y = f(X)$. The standard proof of this fact leaves several questions unanswered. This note explains what goes wrong when attempting a "direct" proof. It also explains how the standard proof... | |
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mathematicaloddsandends.wordpress.com
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| | | | | I recently learned of Craig's formula for the Gaussian Q-function from this blog post from John Cook. Here is the formula: Proposition (Craig's formula). Let $latex Z$ be a standard normal random variable. Then for any $latex z \geq 0$, defining $latex \begin{aligned} \mathbb{P}\{ Z \geq z\} = Q(z) = \dfrac{1}{\sqrt{2\pi}} \int_z^\infty \exp \left( -... | |
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mikespivey.wordpress.com
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| | | | | Equations of the form $latex x^3 = y^2 + k$ are called Mordell equations. In this post we're going to prove that the equation $latex x^3 = y^2 -7$ has no integer solutions, using (with one exception) nothing more complicated than congruences. Theorem: There are no integer solutions to the equation $latex x^3 = y^2... | |
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lucatrevisan.wordpress.com
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| | | We now discuss how to view proofs of certain regularity lemmas as applications of the FTRL methodology. The Szemeredi Regularity Lemma states (in modern language) that every dense graph is well approximate by a graph with a very simple structure, made of the (edge-disjoint) union of a constant number of weighted complete bipartite subgraphs. The... | ||
