IMC附录A

目录

A.1 恒等式与不等式

THEOREM A.1 (Binomial expansion theorem)

PROPOSITION A.2 

PROPOSITION A.3 

PROPOSITION A.4

A.2 渐进符号

DEFINITION A.5

A.3 概率论基础

PROPOSITION A.7 (Union Bound)

THEOREM A.8 (Bayes’ Theorem)

PROPOSITION A.9 

PROPOSITION A.10 (Markov’s inequality)

PROPOSITION A.11 (Chebyshev’s inequality)

COROLLARY A.12

PROPOSITION A.13

PROPOSITION A.14 (Chernoff bound)

A.4 “生日”问题

LEMMA A.15

PROPOSITION A.16

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A.1 恒等式与不等式

THEOREM A.1 (Binomial expansion theorem)

二项式定理

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PROPOSITION A.2 

简单证明一下

 

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PROPOSITION A.3 

切线放缩

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PROPOSITION A.4

For all x with 0 ≤ x ≤ 1 it holds that

可以看作是割线放缩

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A.2 渐进符号

我们使用标准符号来表达函数的渐近行为

DEFINITION A.5

Let f(n), g(n) be functions from non-negative integers to non-negative reals. Then:

通过以下表格可以方便理解和记忆

f(n) = O(g(n))	渐进上界	f(n)的阶数 ≤ g(n)的阶数
f(n) = Ω(g(n))	渐进下界	f(n)的阶数 ≥ g(n)的阶数
f(n) = Θ(g(n))	渐进紧确界	f(n)的阶数 =  g(n)的阶数
f(n) = o(g(n))	非渐进紧确上界	f(n)的阶数 <  g(n)的阶数
f(n) = ω(g(n))	非渐进紧确下界	f(n)的阶数 >  g(n)的阶数

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A.3 概率论基础

PROPOSITION A.7 (Union Bound)

Pr[E1 ∨ E2] ≤ Pr[E1] + Pr[E2]

推广到一般情形，我们有

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THEOREM A.8 (Bayes’ Theorem)

贝叶斯定理

如果Pr[E2]≠0，我们将得到

 

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PROPOSITION A.9 

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PROPOSITION A.10 (Markov’s inequality)

马尔可夫不等式

Let X be a non-negative random variable and v > 0

Pr[X ≥ v] ≤ Exp[X]/v

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PROPOSITION A.11 (Chebyshev’s inequality)

切比雪夫不等式

Let X be a random variable and δ > 0

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COROLLARY A.12

Let X1, . . . , Xm be pairwise-independent random variables with the same expectation µ and variance σ^2.

Then for every δ > 0

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PROPOSITION A.13

 

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PROPOSITION A.14 (Chernoff bound)

切尔诺夫界

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A.4 “生日”问题

LEMMA A.15

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PROPOSITION A.16

若F是伪随机置换，且，则F也是伪随机函数

略去A.5部分