Tail Inequalities 497 - Randomized Algorithms
نویسندگان
چکیده
" Wir mssen wissen, wir werden wissen " (We must know, we shall know) Pr[Y ≥ ∆] = Pr[exp(tY) ≥ exp(t∆)] ≤ E [exp(tY)] exp(t∆) , the first part follows by the fact that exp(·) preserve ordering, and the second part follows by the Markov inequality. Observe that E [exp(tX i)] = 1 2 e t + 1 2 e −t = e t + e −t 2 = 1 2 1 + t 1! + t 2 2! + t 3 3! + · · · + 1 2 1 − t 1! + t 2 2! − t 3 3! + · · · = 1 + + t 2 2! + + · · · + t 2k (2k)! + · · · , by the Taylor expansion of exp(·). Note, that (2k)! ≥ (k!)2 k , and thus
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