Yearly Archives: 2015

Turn Off Voice Assistant on Apple Mac

VoiceOver, as a voice assistant, provides spoken description of items on the screen. The following are two ways of turning on/off VoiceOver. Toggle VoiceOver Use ⌘ Cmd + F5 to toggle on/off VoiceOver. Enable/disable VoiceOver from the menu Open System

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Posted in HowTo

A Double Sum of Binomials

Prove the following summation of binomial coefficients. \[ \sum_{i=0}^n \sum_{j=0}^m {i+j \choose j} = {n + m +2 \choose n+1} - 1. \] First, by Pascal’s rule that \({i+j \choose j} = {i+j-1 \choose j} + {i+j-1 \choose j-1} \),

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Posted in Combinatorics

Chernoff Bound

Consider the sum of a sequence of independent random variables. One may expect its distribution is concentrated around its expected value. This is characterized by the Chernoff bound, which gives exponentially decreasing bounds on tail distributions of the sum. Let

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Posted in Probability