1. 伯努利分布

The experiment has two results,发生概率为$p$、$1-p$,Like tossing a coin.

$X$为离散变量, $X$ ~ $Bernoulli(p)$ ,$P(X=1)=p$

2. 二项分布

进行$n$An independent Bernoulli experiment,成功的次数.such as throwingnThe number of times the coin appears heads,There are times when the touch ball is put back and the black ball is touched.

related to the Bernoulli distribution：The sum of events from the Bernoulli distribution.

$X$~$Binomial(n,p)$

$P(X=x)=C_n^x{p}^x(1-p{)}^{n-x}$

3. 超几何分布

二项分布：There are times when the touch ball is put back and the black ball is touched.
超几何分布：The number of times the black ball was touched without returning the touch ball.

N个球中有M个黑球,Do not put back touch from itn个球,The number of black balls in it.
$P(X=x)=\frac{C_M^x{C}_{N-M}^{n-x}}{C_N^n}$

4. 泊松分布

一定时间/The number of times an event occurs in a space.如1The number of customers arriving in an hour、Number of babies born、The number of cars passing by in an afternoon、The number of misprints for one page of the book.（可根据时间/空间细分,The subdivided events are independent）

$P(N(t)=n)=(ut{)}^n{e}^{-ut}/n!={\lambda }^n{e}^{-\lambda }/n!$
（$u$is the frequency of events within a unit event）

Binomial to Poisson distribution：
Consider the number of vehicles passing through the intersection over a period of time$n$的概率,If the time is subdivided into several small segments$\Delta t$,can be converted to a binomial distribution：每个$\Delta t$There are two possibilities A car passed by/No cars passed by,The probability of a car passing by is $p$.to achieve this condition,Requires infinite subdivisions,即$\Delta t$趋于0,此时$n$趋于无限大,$p$tends to be infinitely small.

即当$n$趋于无限大,$p$tends to infinite hours,The binomial distribution approximates the Poisson distribution.

$P(X=x)=C_n^x{p}^x(1-p{)}^{n-x}\approx {\lambda }^x{e}^{-\lambda }/x!$（其中$\lambda =np$）

5. 几何分布、负二项分布

几何分布：伯努利实验中,The number of heads in the first time

$P(X=x)=(1-p{)}^{x-1}p$

负二项分布：出现r次失败时,成功的次数

6. 指数分布

The waiting time before something happens.

$P(X=t)=\lambda e^{-\lambda t}$

Exponential and Poisson distributions：

$P(X<t)=1-P(N(t)=0)=1-e^{-\lambda t}$

Exponential and geometric distributions：
将时间TInfinitely subdivided inton个时间段,the geometric distribution $p$ 替换为$\lambda T/n$,令$n$趋近于无穷.

echoes the relationship between the binomial distribution and the geometric distribution,泊松分布是“How many times the event occurred in a given time”,The exponential distribution is“How much time has passed until the event happened”.Given an event whose number of occurrences follows a Poisson distribution over a certain period of time,Then the event interval time follows the parameterλThe same exponential distribution

参考文献

https://zhuanlan.zhihu.com/p/47609519