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