随机图论的概率基础

用的是Random Graphs by Béla Bollobás这本书

目录

• Generic Chernoff bounds
• 概率论中的伯恩斯坦不等式
• 概率论中的马尔科夫不等式
• 概率论中的切比雪夫不等式
• the total variation distance
• r-th factorial moment有什么用
• 各种分布之间的联系
• geometric distribution 几何分布
• 负二项分布
• 几何分布的连续版本是指数分布(或负指数分布)
• 超几何分布 从\(N\)个红蓝双色球中抽取\(n\)个球的颜色统计
• 泊松分布
• 更新点Erdős–Rényi graph的东西

几乎可以作为任何需要基础概率论知识的学科的前导资料

Random Graphs by Béla Bollobás 书里给出的就是快问快答的形式，这里摘几个较新鲜的。不定期更新

Generic Chernoff bounds

对于正数\(t\)，

\[{\displaystyle \operatorname {P} \left(X\geq a\right)=\operatorname {P} \left(e^{tX}\geq e^{ta}\right)\leq M(t)e^{-ta}\qquad (t>0)} \]

其中 \(M(t)=\mathbb{E} (e^{tX})\)是随机变量\(X\)的矩生成函数。

概率论中的伯恩斯坦不等式

Bernstein inequalities对随机变量之和偏离其平均值的概率给出了界限。

在最简单的情况下，设\(X_1,\cdots, X_n\)是独立的伯努利随机变量，取值为+1和-1，概率为1/2，则对于每个正的\(\varepsilon\)

\[{\displaystyle \mathbb {P} \left(\left|{\frac {1}{n}}\sum _{i=1}^{n}X_{i}\right|>\varepsilon \right)\leq 2\exp \left(-{\frac {n\varepsilon ^{2}}{2(1+{\frac {\varepsilon }{3}})}}\right).} \]

概率论中的马尔科夫不等式

if \(X\) is a non-negative r.v. with mean \(\mu\) and \(t\geq0\),then

\[\mu\geq P(X\geq t\mu)t\mu \]

改写一下就成为Markov's inequality

\[P(X\geq t\mu)\leq1/t \]

变换一下形式，也是

\[P(Y \geq y )\leq \frac{\mathbb{E}[Y]}{y} \]

概率论中的切比雪夫不等式

Now let \(X\) be a real-valued r.v. with mean \(\mu\) and variance \(\sigma^2\) .if \(d\geq 0\)

\[E\{(X-\mu)^2\}>P(|X-\mu|\geq d)\cdot d^2 \]

改写一下就成为Chebyshev's inequality

\[P(|X-\mu|^2\geq d)\leq \sigma^2/d^2 \]

the total variation distance

Write \(\mathscr{L}(X)\) for the distribution (law) of a r.v. \(X\). Given integer-valued r.vs \(X\) and \(Y\), the total variation distance of \(\mathscr{L}(X)\) and \(\mathscr{L}(Y)\) is

\[d(\mathscr{L}(X), \mathscr{L}(Y))=\sup \{|P(X \in A)-P(Y \in A)|: A \subset \mathbb{Z}\} \]

With a sligt abuse of notation occasionally we write \(d(X,Y)\) or \(d(X,\mathscr{L} (Y))\) instead of \(d(\mathscr{L} (X), \mathscr{L} (Y))\)

r-th factorial moment有什么用

\[E_r(X)=\sum\limits_{k=r}^{\infty}p_k\cdot(k)_r \]

其中\((k)_r\)是下降乘，共\(r\)项

\[(k)_r = k(k-1). ... (k-r+ 1). \]

Note that if \(X\) denotes the number of objects in a certain class then \(E_r(X)\) is the expected number of ordered r-tuples of elements of that class.

各种分布之间的联系

给个链接

http://www.math.wm.edu/~leemis/chart/UDR/UDR.html

geometric distribution 几何分布

The binomial distribution describes the number of successes among n trials, with the probability of a success being p. Now consider the number of failures encountered prior to the first success, and denote this by Y.

\[P(Y=k)=q^kp \ ,k=0,1,... \]

期望\(q/p\),方差\(q/p^2\),r-th factorial moment \(r!(q/p)^r\)

负二项分布

The number of failures prior to the rth success, say \(Zr\), is said to have a negative binomial distribution

\[P(Z_r=k)=\tbinom{r+k-1}{k}p^rq^k \ ,k=0,1,... \]

Since Zr is the sum of r independent geometric r.vs,

期望\(rp/q\),方差\(rq/p^2\)

几何分布的连续版本是指数分布(或负指数分布)

一个非负实随机变量\(L\)被认为具有参数\(\lambda> 0\)的指数分布如果

\[P(L<t)=1-e^{-\lambda t} \ \ for \ t>0 \]

PDF是\(\lambda e^{-\lambda t}\) 期望\(1/\lambda\) 方差\(1/\lambda^2\)

超几何分布 从\(N\)个红蓝双色球中抽取\(n\)个球的颜色统计

The hypergeometric distribution with parameters \(N,R\)and \(n\)\((0<n<N,0<R<N)\)

\[\begin{aligned} q_{k} &=P(X=k)=\left(\begin{array}{l} R \\ k \end{array}\right)\left(\begin{array}{c} N-R \\ n-k \end{array}\right) /\left(\begin{array}{l} N \\ n \end{array}\right) \\ &=\left(\begin{array}{l} n \\ k \end{array}\right)\left(\begin{array}{c} N-n \\ R-k \end{array}\right) /\left(\begin{array}{c} N \\ R \end{array}\right), \quad k=0, \ldots, s \end{aligned} \]

其中\(s=min\{n,R\}\)

泊松分布

\[P(Y=k)=p(k ; \lambda)=\mathrm{e}^{-\lambda} \lambda^{k} / k !, k=0,1, \ldots \]

期望\(\lambda>0\)

更新点Erdős–Rényi graph的东西

这里扔几个链接
https://en.wikipedia.org/wiki/Erdős–Rényi_model
Exact probability of random graph being connected
Probability of not having a path between two certain nodes, in a random graph
Prove that: Probability of connectivity of a random graph is increasing with the size of the graph