隨機圖論的概率基礎

目錄

• 概率論中的馬爾科夫不等式
• 概率論中的切比雪夫不等式
• the total variation distance
• r-th factorial moment有什么用
• 各種分布之間的聯系
• geometric distribution 幾何分布
• 負二項分布
• 幾何分布的連續版本是指數分布(或負指數分布)
• 超幾何分布 從$N$個紅藍雙色球中抽取$n$個球的顏色統計
• 泊松分布
• 更新點Erdős–Rényi graph的東西

幾乎可以作為任何需要基礎概率論知識的學科的前導資料

Random Graphs by Béla Bollobás 書里給出的就是快問快答的形式，這里摘幾個較新鮮的。不定期更新

概率論中的馬爾科夫不等式

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

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

改寫一下就成為Markov's inequality

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

概率論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 \(d\geq 0\)

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

改寫一下就成為Chebyshev's inequality

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

the total variation distance

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\(r\)項

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

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.

各種分布之間的聯系

給個鏈接

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

geometric distribution 幾何分布

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 \(r!(q/p)^r\)

負二項分布

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\(rq/p^2\)

幾何分布的連續版本是指數分布(或負指數分布)

一個非負實隨機變量\(L\)被認為具有參數\(\lambda> 0\)的指數分布如果

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

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

超幾何分布 從\(N\)個紅藍雙色球中抽取\(n\)個球的顏色統計

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\(\lambda>0\)

更新點Erdős–Rényi graph的東西

這里扔幾個鏈接
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