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# System prepended metadata

title: 機率CH5 - Special Discrete Distributions

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# 機率CH5 - Special Discrete Distributions
## 5.1&emsp;Bernoulli and Binomial Random Variables
## 白努力實驗 - 定義probability mass function
- [ ] read
![](https://i.imgur.com/0FD45Y4.png)
## 白努力的期望值、變異數
- [ ] read
![](https://i.imgur.com/sSeS8vi.png)
## Binomial Random Variables
獨立事件、X參數只有n(數量).p(機率數值)
- [ ] read
![](https://i.imgur.com/PqbKWQ9.png)
## 二項分布
- [ ] read
![](https://i.imgur.com/zJ0Uzfy.png)
從n個選x個成功的，順序並不重要
下圖公式裡面還需要多乘一個nCx，投影片錯誤
- [ ] read
![](https://i.imgur.com/Q6G8lJV.png)
## Ex5.8
- [ ] read
![](https://i.imgur.com/OkiMwIR.png)
- [ ] read
![](https://i.imgur.com/UotfdQ2.png)
## Maximum of p(x)
- [ ] read
ceiling function, floor function
![](https://i.imgur.com/qxytDGA.png)
![](https://i.imgur.com/l8nJFfG.png)
![](https://i.imgur.com/6K6f89K.png)
## Binomial distribution的標準差、變異數
- [ ] read
![](https://i.imgur.com/yDTrFuL.png)
![](https://i.imgur.com/OVNIi0V.png)
## 5.2&emsp;Poisson Random Variables
## poisson random variables definition
- [ ] read
![](https://i.imgur.com/IfglBSS.png)
![](https://i.imgur.com/2aRzFWC.png)
![](https://i.imgur.com/xOllEac.png)
![](https://i.imgur.com/Qmilimf.png)
![](https://i.imgur.com/sv7kYDs.png)
![](https://i.imgur.com/5siILp0.png)
## poisson process
- [ ] read
![](https://i.imgur.com/1HZKB7k.png)
Stationary（靜止）找到長度相同的區間，這樣的兩段分佈裡面的數量會相同。區間可以重疊
![](https://i.imgur.com/zr3zbWS.png)
Independent increment 區間不能重疊
![](https://i.imgur.com/3m3TYnD.png)
Orderliness o(h) -> f(h)趨近0的速度比h的還來得快
![](https://i.imgur.com/pb3yXM2.png)
## Th5.2
- [ ] read
![](https://i.imgur.com/XQ8ySBw.png)
proof of th5.2
![](https://i.imgur.com/jJPLtxn.png)
![](https://i.imgur.com/CUpsBMD.png)
![](https://i.imgur.com/WyqNqad.png)
binomial會在極限之下形成poisson
## poisson process with rate
- [ ] read
![](https://i.imgur.com/5NUOGv0.png)
![](https://i.imgur.com/Q59Yqqt.png)
## 5.3&emsp;Other discrete random variables
## Geometric random variables
- [ ] read
![](https://i.imgur.com/q42thPE.png)
## Definition of Geometric random variables
- [ ] read
![](https://i.imgur.com/t1CePok.png)
## Expectation and Variances of Geometric random variables
- [ ] read
![](https://i.imgur.com/3zWvyu2.png)
![](https://i.imgur.com/vmXNW1N.png)
## Memoryless property
- [ ] read
![](https://i.imgur.com/ctoHDwQ.png)
## Geometric random variables is the only Memoryless discrete random variables
- [ ] read
![](https://i.imgur.com/JwAYie7.png)
## Negative Binomial Distribution
- [ ] read
![](https://i.imgur.com/C8JgeLO.png)
![](https://i.imgur.com/bhC4HNA.png)

## With and without replacement
- [ ] read
![](https://i.imgur.com/U8nfjUL.png)

## Hyper-geometric Random Variables 
- [ ] read
![](https://i.imgur.com/wJ05Gid.png)