Randomness in Computing L ECTURE 13 Last time • Finished routing on hypercube • Balls into bins model Today • Poisson distribution • Poisson approximation 3/5/2020 Sofya Raskhodnikova;Randomness in Computing
The number of empty bins 𝒏 balls into 𝒐 bins • The probability that bin 1 is empty is for 𝒚 ≤ 𝟐/𝟑 𝒇 −𝒚−𝒚 𝟑 ≤ 𝟐 − 𝒚 ≤ 𝒚 −𝒚 • Expected number of empty bins 𝑌 = the number of empty bins 𝑌 𝑗 = 𝔽 𝑌 = Sofya Raskhodnikova; Randomness in Computing 3/5/2020
The number of bins with 𝒔 balls 𝒏 balls into 𝒐 bins, 𝒔 is a small constant • The probability 𝑞 𝑠 that bin 1 has 𝑠 balls is 𝑞 𝑠 = Sofya Raskhodnikova; Randomness in Computing 3/5/2020
Poisson random variables • A Poisson random variable with parameters 𝜈 is given by the following distribution on 𝑘 = 0,1,2, … Pr 𝑌 = 𝑘 = 𝑓 −𝜈 𝜈 𝑘 𝑘! 𝒚 𝒌 Taylor expansion: 𝒇 𝒚 = 𝒌=𝟏 • Check that probabilities sum to 1: ∞ 𝒌! 𝑓 −𝜈 𝜈 𝑘 ∞ Pr 𝑌 = 𝑘 = 𝑘=0 ∞ 𝑘=0 = 𝑘! • The expectation of a Poisson R.V. 𝑌 is 𝔽 𝑌 = var 𝑌 = 𝜈 (See Ex. 5.5)
Independent Poisson RVs Theorem Let 𝑌 and 𝑍 be independent Poisson RVs with means 𝜈 𝑌 and 𝜈 𝑍 . Then 𝑌 + 𝑍 is a Poisson RV with mean 𝜈 𝑌 + 𝜈 𝑍 . Sofya Raskhodnikova; Randomness in Computing 3/5/2020
Chernoff Bounds for Poisson RVs Theorem. Let 𝑌 be a Poisson RV with mean 𝜈. • (upper tail, additive) If 𝑦 > 0 , then Pr 𝑌 ≥ 𝜈 + 𝑦 ≤ 𝑓 −𝜈 𝑓𝜈 𝑦 . 𝑦 𝑦 • (lower tail, additive) If 𝑦 < 𝜈 , then Pr 𝑌 ≤ 𝑦 ≤ 𝑓 −𝜈 𝑓𝜈 𝑦 . 𝑦 𝑦 • (upper tail, multiplicative) For any 𝜀 > 0, 𝜈 𝑓 𝜀 Pr 𝑌 ≥ 1 + 𝜀 𝜈 ≤ . 1 + 𝜀 1+𝜀 • (lower tail, multiplicative) For any 𝜀 ∈ (0,1), 𝜈 𝑓 𝜀 Pr 𝑌 ≤ 1 − 𝜀 𝜈 ≤ . 1 − 𝜀 1−𝜀 Sofya Raskhodnikova; Randomness in Computing 3/5/2020
Poisson Distribution is Limit of Binomial Distribution Theorem Let 𝑌 𝑜 ∼ Bin 𝑜, 𝑞 , where 𝑞 is a function of 𝑜 and lim 𝑜→∞ 𝑜𝑞 = 𝜈, a constant independent of 𝑜. Then, for all fixed 𝑙 , 𝑜→∞ Pr[𝑌 𝑜 = 𝑙] = 𝑓 −𝜈 𝜈 𝑙 lim . 𝑙! • Applies to balls-and-bins fi 𝑛 = 𝑜𝑑. Sofya Raskhodnikova; Randomness in Computing 3/5/2020
The Poisson Approximation • The Balls-and-Bins model has dependences. • E.g. if Bin 1 is empty, then Bin 2 is less likely to be empty. • The Poisson Approximation gets rid of dependencies. • (on the board). Sofya Raskhodnikova; Randomness in Computing 3/5/2020
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